# 11-1 Areas of Parallelograms and Triangles. Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary.

Save this PDF as:

Size: px
Start display at page:

Download "11-1 Areas of Parallelograms and Triangles. Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary."

## Transcription

1 Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary Use the Pythagorean Theorem to find the height h, of the parallelogram. Each pair of opposite sides of a parallelogram is congruent to each other. esolutions Manual - Powered by Cognero Page 1

2 3. Use the triangle to find the height. 4. Use the triangle to find the height. Each pair of opposite sides of a parallelogram is congruent to each other. esolutions Manual - Powered by Cognero Page 2

3 5. 6. Use the Pythagorean Theorem to find the height h, of the triangle. Use the Pythagorean Theorem to find the length of the third side of the triangle. Therefore, the perimeter is = 43.5 in. Use the Pythagorean Theorem to find the length of the third side of the triangle. Therefore, the perimeter is = 80 mm. esolutions Manual - Powered by Cognero Page 3

4 7. CRAFTS Marquez and Victoria are making pinwheels. Each pinwheel is composed of 4 triangles with the dimensions shown. Find the perimeter and area of one triangle. 8. Find x. The perimeter is or 28.5 in. Use the Pythagorean Theorem to find the height h, of the triangle. 17 in cm esolutions Manual - Powered by Cognero Page 4

5 CCSS STRUCTURE Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary Use the Pythagorean Theorem to find the height h, of the parallelogram. esolutions Manual - Powered by Cognero Page 5

6 Use the Pythagorean Theorem to find the height h of the triangle. Use the Pythagorean Theorem to find the length of the third side of the triangle. The perimeter is about or 80 mm. Use the Pythagorean Theorem to find the length of the third side of the triangle. The perimeter is about = 69.9 m. esolutions Manual - Powered by Cognero Page 6

7 Use the Pythagorean Theorem to find the length of the other pair of opposite sides of the parallelogram. Use the Pythagorean Theorem to find the length of the other pair of opposite sides of the parallelogram. The perimeter is 2( ) = 170 The perimeter is about 2( ) = TANGRAMS The tangram shown is a 4-inch square. a. Find the perimeter and area of the purple triangle. Round to the nearest tenth. b. Find the perimeter and area of the blue parallelogram. Round to the nearest tenth. a. esolutions Manual - Powered by Cognero Page 7

8 CCSS STRUCTURE Find the area of each parallelogram. Round to the nearest tenth if necessary. Use the Pythagorean Theorem to find the length of the two congruent sides of the triangle. Note that each side is also a hypotenuse for a triangle with sides of length Use the triangle to find the height. The perimeter is about or 9.7 in. b. Use the Pythagorean Theorem to find the length of the other pair of opposite sides of the parallelogram. The perimeter is about 2( ) = 6.8. a. b. esolutions Manual - Powered by Cognero Page 8

9 18. Use the triangle to find the height. 19. Use the triangle to find the height. esolutions Manual - Powered by Cognero Page 9

10 20. Use the triangle to find the height. 22. Use the tangent ratio of an angle to find the height of the parallelogram. 21. esolutions Manual - Powered by Cognero Page 10

11 23. WEATHER Tornado watch areas are often shown on weather maps using parallelograms. What is the area of the region affected by the tornado watch shown? Round to the nearest square mile. 24. The height of a parallelogram is 4 millimeters more than its base. If the area of the parallelogram is 221 square millimeters, find its base and height. The area of a parallelogram is the product of its base length b and its height h. If b = the base, then h = b + 4. Set up and solve the equation for b. We have a figure as shown. Use the cosine ratio of an angle to find the height of the parallelogram. Length cannot be negative. So, b = 13. Therefore, the base of the parallelogram is 13 mm and the height is 17 mm. The area of a parallelogram is the product of its base length b and its height h. b = 13 mm; h = 17 mm b = 394 and h 142. So, the area of the parallelogram is about 394(142) = 55,948. Therefore, an area of about 55,948 mi 2 is affected by the tornado. esolutions Manual - Powered by Cognero Page 11

12 25. The height of a parallelogram is one fourth of its base. If the area of the parallelogram is 36 square centimeters, find its base and height. The area of a parallelogram is the product of its base length b and its height h. 27. The height of a triangle is 3 meters less than its base. If the area of the triangle is 44 square meters, find its base and height. The area of a triangle is half the product of its base length b and its height h. Solve the equation for x. Length cannot be negative, so b = 12 and h = 3. Therefore, the base of the parallelogram is 12 cm and the height is 3 cm. b = 12 cm; h = 3 cm 26. The base of a triangle is twice its height. If the area of the triangle is 49 square feet, find its base and height. b = 11 or 8. Length cannot be negative, so b = 11. Therefore, the base of the triangle is 11 m and the height is 8 m. b = 11 m; h = 8 m The area of a triangle is half the product of its base length b and its height h. Length cannot be negative, so the height is 7 and the base is 14. b = 14 ft; h = 7 ft esolutions Manual - Powered by Cognero Page 12

13 28. FLAGS Omar wants to make a replica of Guyana s national flag. 29. DRAMA Madison is in charge of the set design for her high school s rendition of Romeo and Juliet. One pint of paint covers 80 square feet. How many pints will she need of each color if the roof and tower each need 3 coats of paint? a. What is the area of the piece of fabric he will need for the red region? for the yellow region? b. If the fabric costs \$3.99 per square yard for each color and he buys exactly the amount of fabric he needs, how much will it cost to make the flag? a. The red region is a triangle with a base of 2 feet and a height of 1 foot. The area of a triangle is half the product of a base b and its corresponding height h. So, the area of the fabric required for the red region is The area of the yellow region is the difference between the areas of the triangle of base 2 ft and height 2 ft and the red region. The area of the triangle of base 2 ft and height 2 ft is and that of the red region is 1 ft 2. Therefore, the fabric required for the yellow region is 2 1 = 1 ft 2. b. The amount of fabric that is required for the entire flag is 2 ft 2 ft or 4 ft 2. The area of the roof is equal to 2.5 ft 6 ft or 15 ft 2. If 3 coats of paint are needed, 15 3 or 45 square feet need to be painted. One pint of yellow paint can cover 80 square feet, so only 1 pint of yellow paint is needed. The area of the tower is equal to 12 ft 5 ft or 60 ft 2. If 3 coats of paint are needed, 60 3 or 180 square feet need to be painted pints of blue paint will be needed, so she will need to purchase 3 pints. 1 pint of yellow, 3 pints of blue Therefore, the total cost to make the flag will be about \$1.77. a. b. \$1.77 esolutions Manual - Powered by Cognero Page 13

14 Find the perimeter and area of each figure. Round to the nearest hundredth, if necessary. 30. Use the triangle to find the lengths of the sides. 31. Use trigonometry to find the width of the rectangle. The perimeter is 4 2 = 8 The area is 2 2 or 4 m 2. 8 m; 4 m in.; 4.79 in Use trigonometry to find the side length of the parallelogram. esolutions Manual - Powered by Cognero Page 14

15 COORDINATE GEOMETRY Find the area of each figure. Explain the method that you used. 33. ABCD with A(4, 7), B(2, 1), C(8, 1), and D(10, 7) Graph the diagram. Use trigonometry to find h. The base of the parallelogram is horizontal and from x = 2 to x = 8, so it is 6 units long. The height of the parallelogram vertical and from y = 1 to y = 7, so it is 6 units long. The area is (6)(6) = 36 units 2. ; Graph the parallelogram, then measure the length of the base and the height and calculate the area yd; yd 2 esolutions Manual - Powered by Cognero Page 15

16 34. RST with R( 8, 2), S( 2, 2), and T( 3, 7) Graph the diagram. The base of the triangle is horizontal and from x = 2 to x = 8, so it is 6 units long. 35. HERON S FORMULA Heron s Formula relates the lengths of the sides of a triangle to the area of the triangle. The formula is, where s is the semiperimeter, or one half the perimeter, of the triangle and a, b, and c are the side lengths. a. Use Heron s Formula to find the area of a triangle with side lengths 7, 10, and 4. b. Show that the areas found for a right triangle are the same using Heron s Formula and using the triangle area formula you learned earlier in this lesson. a. Substitute a = 7, b = 10 and c = 4 in the formula. The height of the triangle vertical and from y = 2 to y = 7, so it is 5 units long. The area is 0.5(6)(5) = 15 units 2. ; Graph the triangle, then measure the length of the base and the height and calculate the area. b. a. b. 36. MULTIPLE REPRESENTATIONS In this problem, you will investigate the relationship between esolutions Manual - Powered by Cognero Page 16

17 the area and perimeter of a rectangle. a. ALGEBRAIC A rectangle has a perimeter of 12 units. If the length of the rectangle is x and the width of the rectangle is y, write equations for the perimeter and area of the rectangle. b. TABULAR Tabulate all possible whole-number values for the length and width of the rectangle, and find the area for each pair. c. GRAPHICAL Graph the area of the rectangle with respect to its length. d. VERBAL Describe how the area of the rectangle changes as its length changes. e. ANALYTICAL For what whole-number values of length and width will the area be greatest? least? Explain your reasoning. a. P = 2x + 2y; A = xy b. Solve for y. Now input values for x to get corresponding values for y. Find xy to get the area. Area is inputted as the y-coordinates. d. Sample answer: The plots (and the y-coordinates) go up as the x-coordinates move from 1 to 3, and then move down as the x-coordinates move from 3 to 5. The area increases as the length increases from 1 to 3, is highest at 3, then decreases as the length increases to 5. e. Sample answer: The graph reaches its highest point when x = 3, so the area of the rectangle will be greatest when the length is 3. The graph reaches its lowest points when x = 1 and 5, so the area of the rectangle will be the smallest when the length is 1 or 5, assuming the lengths are always whole numbers. a. P = 2x + 2y; A = xy b. c. Length is x and is inputted as the x-coordinates. Area is inputted as the y-coordinates. c. esolutions Manual - Powered by Cognero Page 17

18 c. triangles as shown. d. Sample answer: The area increases as the length increases from 1 to 3, is highest at 3, then decreases as the length increases to 5. e. Sample answer: The graph reaches its highest point when x = 3, so the area of the rectangle will be greatest when the length is 3. The graph reaches its lowest points when x = 1 and 5, so the area of the rectangle will be the smallest when the length is 1 or CHALLENGE Find the area of ABC. Explain your method. The area of the square is 36 units 2. The three new triangles are all right triangles and the lengths of their sides can be found by subtracting the coordinates. The areas of the three triangles are 6 unit 2, 6 unit 2, and 9 unit 2 respectively. Therefore, the area of the main triangle is the difference or 15 unit 2. ; Sample answer: I inscribed the triangle in a 6-by-6 square. I found the area of the square and subtracted the areas of the three right triangles inside the square that were positioned around the given triangle. The area of the given triangle is the difference, or. One method of solving would be to find the length of one of the bases and then calculate the corresponding height. The length of the sides can be found by using the distance formula; for example by finding the distance between A and C to calculate the length of AC. The height will be more challenging, because we will need to determine the perpendicular from point B to line AC. We could also use Heron's formula, which is described in problem #35 in this lesson. A third method will be to inscribe the triangle in a 6 by 6 square. When we do this, we form three new triangles as shown. esolutions Manual - Powered by Cognero Page 18

19 38. CCSS ARGUMENTS Will the perimeter of a nonrectangular parallelogram sometimes, always, or never be greater than the perimeter of a rectangle with the same area and the same height? Explain. Always; sample answer: If the areas are equal, the perimeter of the nonrectangular parallelogram will always be greater because the length of the side that is not perpendicular to the height forms a right triangle with the height. The height is a leg of the triangle and the side of the parallelogram is the hypotenuse of the triangle. Since the hypotenuse is always the longest side of a right triangle, the nonperpendicular side of the parallelogram is always greater than the height. The bases of the quadrilaterals have to be the same because the areas and the heights are the same. Since the bases are the same and the height of the rectangle is also the length of a side, the perimeter of the parallelogram will always be greater. 39. WRITING IN MATH Points J and L lie on line m. K lies on line p. If lines m and p are parallel, describe the area of JKL will change as K moves along line Sample answer: The area will not change as K move along line p. Since lines m and p are parallel, the perpendicular distance between them is constant. Th means that no matter where K is on line p, the perpendicular distance to line p, or the height of the triangle, is always the same. Always; sample answer: If the areas are equal, the perimeter of the nonrectangular parallelogram will always be greater because the length of the side that is not perpendicular to the height forms a right triangle with the height. The height is a leg of the triangle and the side of the parallelogram is the hypotenuse of the triangle. Since the hypotenuse is always the longest side of a right triangle, the nonperpendicular side of the parallelogram is always greater than the height. The bases of the quadrilaterals have to be the same because the areas and the heights are the same. Since the bases are the same and the height of the rectangle is also the length of a side, the perimeter of the parallelogram will always be greater. Since point J and L are not moving, the distance bet them, or the length of the base, is constant. Since the height of the triangle and the base of the triangle are constant, the area will always be the same. Sample answer: The area will not change as K move along line p. Since lines m and p are parallel, the perpendicular distance between them is constant. Th means that no matter where K is on line p, the perpendicular distance to line p, or the height of the triangle, is always the same. Since point J and L are moving, the distance between them, or the length of t base, is constant. Since the height of the triangle and base of the triangle are both constant, the area will al be the same. 40. OPEN ENDED The area of a polygon is 35 square units. The height is 7 units. Draw three different triangles and three different parallelograms that meet these requirements. Label the base and height on each. For each triangle, maintain a height of 7 and a base of 10, and the area will not change from 35 square units. For each parallelogram, maintain a height of 7 and a base of 5, and the area will not change from 35 esolutions Manual - Powered by Cognero Page 19

20 square units. parallelogram PQRS. Sample answer: The area of a parallelogram is the product of the base and the height, where the height is perpendicular to the base. Opposite sides of a parallelogram are parallel so either base can be multiplied by the height to find the area. To find the area of the parallelogram, you can measure the height bases to get the area. and then measure one of the and multiply the height by the base You can also measure the height and measure one of the bases and then multiply the height by the base to get the area. 41. WRITING IN MATH Describe two different ways you could use measurement to find the area of parallelogram PQRS. It doesn t matter which side you choose to use as the base, as long as you use the height that is perpendicular to that base to calculate the area. Sample answer: To find the area of the parallelogram, you can measure the height and esolutions Manual - Powered by Cognero Page 20

21 parallelogram, you can measure the height and then measure one of the bases and multiply the height by the base to get the area. You can also measure the height and measure one of the bases and then multiply the height by the base to get the area. It doesn t matter which side you choose to use as the base, as long as you use the height that is perpendicular to that base to calculate the area. 42. What is the area, in square units, of the parallelogram shown? 43. GRIDDED RESPONSE In parallelogram ABCD, intersect at E. If AE = 9, BE = 3x 7, and DE = x + 5, find x. In a parallelogram, diagonals bisect each other. So, BE = DE, and 3x 7 = x A 12 B 20 C 32 D 40 One of the bases along the y-axis from y = 0 to y = 8, so it is 8 units. The height is horizontal and from x = 0 to x = 4, so the height is 4 units. The area is (8)(4) = 32 units 2. The correct choice is C. C esolutions Manual - Powered by Cognero Page 21

22 44. A wheelchair ramp is built that is 20 inches high and has a length of 12 feet as shown. What is the measure of the angle x that the ramp makes with the ground, to the nearest degree? F 8 G 16 H 37 J 53 Use the sine ratio of the angle xº to find the value. Here opposite side is 20 inches and the hypotenuse is 12 ft = 144 in. 45. SAT/ACT The formula for converting a Celsius temperature to a Fahrenheit temperature is, where F is the temperature in degrees Fahrenheit and C is the temperature in degrees Celsius. Which of the following is the Celsius equivalent to a temperature of 86 Fahrenheit? A 15.7 C B 30 C C 65.5 C D C D C Substitute 86 for F in the equation and solve for C. Therefore, the correct choice is B. B Therefore, the correct choice is F. F Write the equation of each circle. 46. center at origin, r = 3 The standard form of the equation of a circle with center at (h, k) and radius r is (x h) 2 + (y k) 2 = r 2. (h, k) = (0, 0) and r = 3. Therefore, the equation is (x 0) 2 + (y 0) 2 = 3 2 or x 2 + y 2 = 9. esolutions Manual - Powered by Cognero Page 22

23 47. center at origin, d = 12 The standard form of the equation of a circle with center at (h, k) and radius r is (x h) 2 + (y k) 2 = r center at (1, 4), The standard form of the equation of a circle with center at (h, k) and radius r is (x h) 2 + (y k) 2 = r 2. (h, k) = (0, 0) and r = d 2 = 6. Therefore, the equation is (x 0) 2 + (y 0) 2 = 6 2. or x 2 + y 2 = 36. (h, k) = (1, 4) and. Therefore, the equation is (x 1) 2 + (y [ 4]) 2 = or (x 1) 2 + (y + 4) 2 = center at ( 3, 10), d = 24 The standard form of the equation of a circle with center at (h, k) and radius r is (x h) 2 + (y k) 2 = r 2. (h, k) = ( 3, 10) and r = d 2 = 12. Therefore, the equation is (x [ 3]) 2 + (y [ 10]) 2 = r 2 or (x + 3) 2 + (y + 10) 2 = 144. Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent. 50. If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment. x = 4 or 9 Since x is a length, it cannot be negative. Therefore, x = 4. 4 esolutions Manual - Powered by Cognero Page 23

24 51. If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment. 52. If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment. Use the Quadratic Formula to find the roots. Since x is a length, it cannot be negative. Therefore, x = OPTIC ART Victor Vasarely created art in the op art style. This piece, AMBIGU-B, consists of multicolored parallelograms. Describe one method to ensure that the shapes are parallelograms. Refer to the photo on Page 770. Sample answer: If each pair of opposite sides are parallel, the quadrilateral is a parallelogram. Sample answer: If each pair of opposite sides are parallel, the quadrilateral is a parallelogram. Evaluate each expression if a = 2, b = 6, and c = Substitute a = 2 and c = 3 in the expression. 3 esolutions Manual - Powered by Cognero Page 24

25 55. Substitute b = 6 and c = 3 in the expression Substitute a = 2, b = 6, and c = 3 in the expression Substitute a = 2, b = 6, and c = 3 in the expression Substitute a = 2, b = 6, and c = 3 in the expression. 12 esolutions Manual - Powered by Cognero Page 25

### Each pair of opposite sides of a parallelogram is congruent to each other.

Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary. 1. Use the Pythagorean Theorem to find the height h, of the parallelogram. 2. Each pair of opposite

### 11-2 Areas of Trapezoids, Rhombi, and Kites. Find the area of each trapezoid, rhombus, or kite. 1. SOLUTION: 2. SOLUTION: 3.

Find the area of each trapezoid, rhombus, or kite. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. OPEN ENDED Suki is doing fashion design at 4-H Club. Her first project is to make a simple A-line

### 11-4 Areas of Regular Polygons and Composite Figures

1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,

### as a fraction and as a decimal to the nearest hundredth.

Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. sin A The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. So, 2. tan C The tangent of an

### Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 42. The sum of the measures of the angles of a triangle is 180. Therefore, The sine of an angle

### 12-2 Surface Areas of Prisms and Cylinders. 1. Find the lateral area of the prism. SOLUTION: ANSWER: in 2

1. Find the lateral area of the prism. 3. The base of the prism is a right triangle with the legs 8 ft and 6 ft long. Use the Pythagorean Theorem to find the length of the hypotenuse of the base. 112.5

### 10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

### 10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: ANSWER: 2 SOLUTION: ANSWER:

Find x. Assume that segments that appear to be tangent are tangent. 1. 3. 2 5 2. 4. 6 13 esolutions Manual - Powered by Cognero Page 1 5. SCIENCE A piece of broken pottery found at an archaeological site

### 8-2 The Pythagorean Theorem and Its Converse. Find x.

Find x. 1. of the hypotenuse. The length of the hypotenuse is 13 and the lengths of the legs are 5 and x. 2. of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 12.

### 10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: 2. SOLUTION: 3.

Find x. Assume that segments that appear to be tangent are tangent. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. 5. SCIENCE A piece of broken pottery found at an archaeological site is shown.

### 12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. SOLUTION: ANSWER: 1017.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 3. sphere: area of great circle = 36π yd 2 We know that the area of a great circle is r.. Find 1. Now find the surface area.

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### (a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

### 7-1 Ratios and Proportions

1. PETS Out of a survey of 1000 households, 460 had at least one dog or cat as a pet. What is the ratio of pet owners to households? The ratio of per owners to households is 23:50. 2. SPORTS Thirty girls

### 12-8 Congruent and Similar Solids

Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. 3. Two similar cylinders have radii of 15 inches and 6 inches. What is the ratio

### are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

For Exercises 1 4, refer to. 1. Name the circle. The center of the circle is N. So, the circle is 2. Identify each. a. a chord b. a diameter c. a radius a. A chord is a segment with endpoints on the circle.

### 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft

2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite

### 11-5 Areas of Similar Figures. For each pair of similar figures, find the area of the green figure. SOLUTION:

The area of the green diamond 9 yd F each pair of similar figures, find the area of the green figure The scale fact between the blue triangle and the 1 green triangle The scale fact between the blue diamond

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### 11-4 Areas of Regular Polygons and Composite Figures

1.In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. SOLUTION: Center: point F, radius:,

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### Copy each polygon and point K. Then use a protractor and ruler to draw the specified rotation of each figure about point K

Copy each polygon and point K. Then use a protractor and ruler to draw the specified rotation of each figure about point K. 1. 45 Step 1: Draw a segment from G to K. Step 2: Draw a 45 angle using. Step

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Practice Test - Chapter 1. Use the figure to name each of the following.

Use the figure to name each of the following. 1. the line that contains points Q and Z The points Q and Z lie on the line b, line b 2. two points that are coplanar with points W, X, and Y Coplanar points

### 12-8 Congruent and Similar Solids

Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. Ratio of radii: Ratio of heights: The ratios of the corresponding measures are

### 6-1 Angles of Polygons

Find the sum of the measures of the interior angles of each convex polygon. 1. decagon A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures.

### 12-4 Volumes of Prisms and Cylinders. Find the volume of each prism.

Find the volume of each prism. 3. the oblique rectangular prism shown at the right 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. If two solids have

### Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

### 12-4 Volumes of Prisms and Cylinders. Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h

Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h The volume is 108 cm 3. The volume V of a prism is V = Bh, where B is the area of a base and h the

### 13-3 Geometric Probability. 1. P(X is on ) SOLUTION: 2. P(X is on ) SOLUTION:

Point X is chosen at random on. Find the probability of each event. 1. P(X is on ) 2. P(X is on ) 3. CARDS In a game of cards, 43 cards are used, including one joker. Four players are each dealt 10 cards

### 11-6 Area: Parallelograms, Triangles, and Trapezoids

1. 6. LACROSSE A lacrosse goal with net is shown. The goal is 6 feet wide, 6 feet high, and 7 feet deep. What is the area of the triangular region of the ground inside the net? 30.5 ft 2 2. 21 ft 2 14.08

### The scale factor between the blue triangle and the green triangle is

For each pair of similar figures, find the area of the green figure 1 SOLUTION: The scale factor between the blue diamond and the green diamond is 2 The area of the green diamond is 9 yd 2 SOLUTION: The

### , where B is the area of the base and h is the height of the pyramid. The base

Find the volume of each pyramid. The volume of a pyramid is, where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5

### 4-6 Inverse Trigonometric Functions

Find the exact value of each expression, if it exists. 1. sin 1 0 Find a point on the unit circle on the interval with a y-coordinate of 0. 3. arcsin Find a point on the unit circle on the interval with

### Right Triangle Trigonometry Test Review

Class: Date: Right Triangle Trigonometry Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the length of the missing side. Leave your answer

### Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given

### MAT 080-Algebra II Applications of Quadratic Equations

MAT 080-Algebra II Applications of Quadratic Equations Objectives a Applications involving rectangles b Applications involving right triangles a Applications involving rectangles One of the common applications

### Skills Practice Simplifying Radical Expressions. Lesson Simplify c 2 d x 4 y m 5 n

11-1 Simplify. Skills Practice Simplifying Radical Expressions 1. 28 2. 40 3. 72 4. 99 5. 2 10 6. 5 60 7. 3 5 5 8. 6 4 24 Lesson 11-1 9. 2 3 3 15 10. 16b 4 11. 81c 2 d 4 12. 40x 4 y 6 13. 75m 5 n 2 5 14.

1. Determine whether each quadrilateral is a Justify your answer. 3. KITES Charmaine is building the kite shown below. She wants to be sure that the string around her frame forms a parallelogram before

### 8-1 Geometric Mean. or Find the geometric mean between each pair of numbers and 20. similar triangles in the figure.

8-1 Geometric Mean or 24.5 Find the geometric mean between each pair of numbers. 1. 5 and 20 4. Write a similarity statement identifying the three similar triangles in the figure. numbers a and b is given

### 6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### An arrangement that shows objects in rows and columns Example:

1. acute angle : An angle that measures less than a right angle (90 ). 2. addend : Any of the numbers that are added 2 + 3 = 5 The addends are 2 and 3. 3. angle : A figure formed by two rays that meet

### 8-2 The Pythagorean Theorem and Its Converse. Find x.

1 8- The Pythagorean Theorem and Its Converse Find x. 1. hypotenuse is 13 and the lengths of the legs are 5 and x.. equaltothesquareofthelengthofthehypotenuse. The length of the hypotenuse is x and the

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### Test to see if ΔFEG is a right triangle.

1. Copy the figure shown, and draw the common tangents. If no common tangent exists, state no common tangent. Every tangent drawn to the small circle will intersect the larger circle in two points. Every

### Geometry, Final Review Packet

Name: Geometry, Final Review Packet I. Vocabulary match each word on the left to its definition on the right. Word Letter Definition Acute angle A. Meeting at a point Angle bisector B. An angle with a

### Name: Class: Date: Geometry Chapter 3 Review

Name: Class: Date: ID: A Geometry Chapter 3 Review. 1. The area of a rectangular field is 6800 square meters. If the width of the field is 80 meters, what is the perimeter of the field? Draw a diagram

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### CK-12 Geometry: Parts of Circles and Tangent Lines

CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.

### Therefore, x is 3. To find BC. find 2x or 2(3) = 6. Thus, BC is Linear Measure

Find the length of each line segment or object. 1. Refer to Page 18. The ruler is marked in centimeters. The tail of the fish starts at the zero mark of the ruler and the mouth appears to end 7 tenth marks

### The area of a figure is the measure of the size of the region enclosed by the figure. Formulas for the area of common figures: square: A = s 2

The area of a figure is the measure of the size of the region enclosed by the figure. Formulas for the area of common figures: square: A = s 2 s s rectangle: A = l w parallelogram: A = b h h b triangle:

### Geometry Review. Here are some formulas and concepts that you will need to review before working on the practice exam.

Geometry Review Here are some formulas and concepts that you will need to review before working on the practice eam. Triangles o Perimeter or the distance around the triangle is found by adding all of

### Archdiocese of Washington Catholic Schools Academic Standards Mathematics

5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,

### Unit 10: Quadratic Equations Chapter Test

Unit 10: Quadratic Equations Chapter Test Part 1: Multiple Choice. Circle the correct answer. 1. The area of a square is 169 cm 2. What is the length of one side of the square? A. 84.5 cm C. 42.25 cm B.

### 6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 3. PROOF Write a two-column proof to prove that if ABCD is a rhombus with diagonal. 1. If, find. A rhombus is a parallelogram with all

### State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. esolutions Manual - Powered by Cognero Page 1 1. A figure

### Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.

Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 1. Use the Law of Cosines to find the missing side length. Use the Law of Sines to find a missing

### II. Geometry and Measurement

II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 17 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### 10.1: Areas of Parallelograms and Triangles

10.1: Areas of Parallelograms and Triangles Important Vocabulary: By the end of this lesson, you should be able to define these terms: Base of a Parallelogram, Altitude of a Parallelogram, Height of a

### Geometry SOL G.11 G.12 Circles Study Guide

Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter-10-circles-flashcardsflash-cards/).

### of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 459-2058 Mobile: (949) 510-8153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:

### Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain

AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent

### Grade 4 Math Expressions Vocabulary Words

Grade 4 Math Expressions Vocabulary Words Link to Math Expression Online Glossary for some definitions: http://wwwk6.thinkcentral.com/content/hsp/math/hspmathmx/na/gr4/se_9780547153131_/eglos sary/eg_popup.html?grade=4

### Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### 1-6 Two-Dimensional Figures. Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular.

Stop signs are constructed in the shape of a polygon with 8 sides of equal length The polygon has 8 sides A polygon with 8 sides is an octagon All sides of the polygon are congruent and all angles are

### ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE

1 ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don t get the answer given, check your work and

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### 12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

1-6 Surface Area and Volumes of Spheres Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 6/87,1 6/87,1 sphere: area of great circle = 36ʌ yd 6/87,1 We know that the area

### Perimeter and Area of Geometric Figures on the Coordinate Plane

Perimeter and Area of Geometric Figures on the Coordinate Plane There are more than 00 national flags in the world. One of the largest is the flag of Brazil flown in Three Powers Plaza in Brasilia. This

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### Geometry FSA Mathematics Practice Test Questions

Geometry FSA Mathematics Practice Test Questions The purpose of these practice test materials is to orient teachers and students to the types of questions on paper-based FSA tests. By using these materials,

### Honors Geometry Final Exam Study Guide

2011-2012 Honors Geometry Final Exam Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In each pair of triangles, parts are congruent as marked.

### Cumulative Test. 161 Holt Geometry. Name Date Class

Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2

### Mid-Chapter Quiz: Lessons 4-1 through 4-4

Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to

### THE DISTANCE FORMULA

THE DISTANCE FORMULA In this activity, you will develop a formula for calculating the distance between any two points in a coordinate plane. Part 1: Distance Along a Horizontal or Vertical Line To find

### 5-6 The Remainder and Factor Theorems

Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 Divide the function by x 4. The remainder is 58. Therefore, f (4) = 58. Divide the function by x + 2.

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

### 10-3 Area of Parallelograms

0-3 Area of Parallelograms MAIN IDEA Find the areas of parallelograms. NYS Core Curriculum 6.A.6 Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest,

### Integrated Algebra: Geometry

Integrated Algebra: Geometry Topics of Study: o Perimeter and Circumference o Area Shaded Area Composite Area o Volume o Surface Area o Relative Error Links to Useful Websites & Videos: o Perimeter and

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### Write the equation using h = 0, k = 0, and r = 10-8 Equations of Circle. Write the equation of each circle. 1. center at (9, 0), radius 5 SOLUTION:

Write the equation of each circle 1center at (9, 0), radius 5 2center at (3, 1), diameter 14 Since the radius is half the diameter, r = (14) or 7 3center at origin, passes through (2, 2) Find the distance

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### Pythagorean Theorem, Distance and Midpoints Chapter Questions. 3. What types of lines do we need to use the distance and midpoint formulas for?

Pythagorean Theorem, Distance and Midpoints Chapter Questions 1. How is the formula for the Pythagorean Theorem derived? 2. What type of triangle uses the Pythagorean Theorem? 3. What types of lines do

### Name: Perimeter and area November 18, 2013

1. How many differently shaped rectangles with whole number sides could have an area of 360? 5. If a rectangle s length and width are both doubled, by what percent is the rectangle s area increased? 2.

### GEOMETRY (Common Core)

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (Common Core) Tuesday, June 2, 2015 1:15 to 4:15 p.m. MODEL RESPONSE SET Table of Contents Question 25...................

### FS Geometry EOC Review

MAFS.912.G-C.1.1 Dilation of a Line: Center on the Line In the figure, points A, B, and C are collinear. http://www.cpalms.org/public/previewresource/preview/72776 1. Graph the images of points A, B, and

### GEOMETRY FINAL EXAM REVIEW

GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.

### Geometry Chapter 9 Extending Perimeter, Circumference, and Area

Geometry Chapter 9 Extending Perimeter, Circumference, and Area Lesson 1 Developing Formulas for Triangles and Quadrilaterals Learning Target (LT-1) Solve problems involving the perimeter and area of triangles

### Geometry Chapter 9 Extending Perimeter, Circumference, and Area

Geometry Chapter 9 Extending Perimeter, Circumference, and Area Lesson 1 Developing Formulas for Triangles and Quadrilaterals Learning Targets LT9-1: Solve problems involving the perimeter and area of

### The Pythagorean Packet Everything Pythagorean Theorem

Name Date The Pythagorean Packet Everything Pythagorean Theorem Directions: Fill in each blank for the right triangle by using the words in the Vocab Bo. A Right Triangle These sides are called the of

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### MATH 139 FINAL EXAM REVIEW PROBLEMS

MTH 139 FINL EXM REVIEW PROLEMS ring a protractor, compass and ruler. Note: This is NOT a practice exam. It is a collection of problems to help you review some of the material for the exam and to practice