10-1. Areas of Parallelograms and Triangles. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

Transcription

1 - Areas of Parallelograms and Triangles Vocabulary Review The diagram below shows the different types of parallelograms. Parallelogram Rhombus Rectangle Square Underline the correct word to complete each sentence.. All parallelograms are quadrilaterals / rectangles.. All parallelograms have opposite sides parallel / perpendicular. 3. Some parallelograms are trapezoids / rectangles. Vocabulary Builder area (noun) EHR ee uh Definition: Area is the number of square units needed to cover a given surface. Main Idea: You can find the area of a parallelogram or a triangle when you know the length of its base and its height. Use Your Vocabulary Find the area of each figure square units 8 square units 4.5 square units hapter 5

2 Theorems - and - Area of a Rectangle and a Parallelogram Theorem - Area of a Rectangle The area of a rectangle is the product of its base and height. A 5 bh Theorem - Area of a Parallelogram The area of a parallelogram is the product of a base and the corresponding height. A 5 bh b h b h 7. Explain how finding the area of a parallelogram and finding the area of a rectangle are alike. Explanations may vary. Sample: For each figure, you find the product of the base and its corresponding height. Problem Finding the Area of a Parallelogram Got It? What is the area of a parallelogram with base length m and height 9 m? 8. Label the parallelogram at the right. 9. Find the area. A 5 bh Problem Write the formula. 5 ( 9 ) Substitute. 5 8 Simplify.. The area of the parallelogram is 8 m. Finding a Missing Dimension 9 cm 9 m m Got It? A parallelogram has sides 5 cm and 8 cm. The height corresponding to a 5-cm base is 9 cm. What is the height corresponding to an 8-cm base?. Label the parallelogram at the right. Let h represent the height corresponding to the 8-cm base.. Find the area. A 5 bh 5 5? h 8 cm 5 cm 3. The area of the parallelogram is 35 cm. 5 Lesson -

3 4. Use the area of the parallelogram to find the height corresponding to an 8-cm base. A 5 bh Write the formula ( 8 )h Substitute ( 8 )h Divide each side by the length of the base h Simplify. 5. The height corresponding to an 8-cm base is 7.5 cm. Theorem -3 Area of a Triangle The area of a triangle is half the product of a base and the corresponding height. h A 5 bh b 6. Explain how finding the area of a triangle is different from finding the area of a rectangle. Explanations may vary. Sample: For a triangle, find half the product of the base and height. For a rectangle, find the product of the base and height. Problem 3 Finding the Area of a Triangle Got It? What is the area of the triangle? 7. ircle the formula you can use to find the area of the triangle. A 5 bh A 5 bh 8. onvert the lengths of the base and the hypotenuse to inches. base hypotenuse ft 5 in. ft in. 5 3 in. 9. Find the area of the triangle. A 5 bh 5 ()(5) 5 (6) in. ft in. ft. The area of the triangle is 3 in.. hapter 5

4 Problem 4 Finding the Area of an Irregular Figure Got It? Reasoning Suppose the base lengths of the square and triangle in the figure are doubled to in., but the height of each polygon remains the same. How is the area of the figure affected? 8 in.. omplete to find the area of each irregular figure. Area of Original Irregular Figure Area of New Irregular Figure A 5 6(6) (6)(8) A 5 ()(6)(6) ()(6)(8) ()(36) () ( 4 ) ()( 36 4 ) 5 () ( 6 ) 5. How is the area affected? The area is doubled. 6 in. Lesson heck Do you UNDERSTAND? ~ABD is divided into two triangles along diagonal A. If you know the area of the parallelogram, how do you find the area of kab? D Write T for true or F for false. T 3. Since A is a diagonal of ~ABD, nab is congruent to nda. A B F 4. The area of nab is greater than the area nda. T Math Success heck off the vocabulary words that you understand. base of a parallelogram base of a triangle Need to review 5. The area of nab is half the area of ~ABD. 6. If you know the area of the parallelogram, how do you find the area of nab? Divide the area of the parallelogram by height of a parallelogram height of a triangle Rate how well you can find the area of parallelograms and triangles. Now I get it! 53 Lesson -

5 - Areas of Trapezoids, Rhombuses, and Kites Vocabulary Review. Is a rhombus a parallelogram? Yes / No. Are all rhombuses squares? Yes / No 3. Are all squares rhombuses? Yes / No 4. ross out the figure that is NOT a rhombus Vocabulary Builder kite (noun) kyt Definition: A kite is a quadrilateral with two pairs of congruent adjacent sides. Main Idea: You can find the area of a kite when you know the lengths of its diagonals. Word Origin: The name for this quadrilateral is taken from the name of the flying toy that it looks like. Use Your Vocabulary 5. ircle the kite The figure at the right is a kite. What is the value of x? Explain. 5; a kite has two pairs of congruent adjacent sides x hapter 54

6 Theorem -4 Area of a Trapezoid The area of a trapezoid is half the product of the height and the sum of the bases. A 5 h(b b ) Underline the correct word to complete each sentence. 7. The bases of a trapezoid are parallel / perpendicular. 8. The height / width of a trapezoid is the perpendicular distance between the bases. h b b Problem Area of a Trapezoid Got It? What is the area of a trapezoid with height 7 cm and bases cm and 5 cm? 9. Use the justifications below to find the area of the trapezoid. A 5 h(b b ) Use the formula for area of a trapezoid. 5 ( 7 )( 5) Substitute. 5 ( 7 )( 7 ) Add Simplify.. The area of the trapezoid is 94.5 cm. Problem Finding Area Using a Right Triangle Got It? Suppose h decreases in trapezoid PQRS so that mlp 5 45 while angles R and Q and the bases stay the same. What is the area of trapezoid PQRS?. If m/p 5 45, is the triangle still a triangle? Yes / No. Is the triangle a triangle? Yes / No 3. Are the legs of a triangle congruent? Yes / No 4. The height of the triangle is m. 5. The area is found below. Write a justification for each step. A 5 h(b b ) 5 ()(5 7) 5 ()() 5 Use the trapezoid area formula. Substitute. Add. Simplify. P S h 6 m 5 m 5 m R Q 6. The area of trapezoid PQRS is m. 55 Lesson -

7 Theorem -5 Area of a Rhombus or a Kite The area of a rhombus or a kite is half the product of the lengths of its diagonals. d d d A 5 (d d ) d Rhombus 7. Describe one way that finding the area of rhombus or a kite is different from finding the area of a trapezoid. Kite Answers may vary. Sample: Instead of using the height and the lengths of the bases, you use the lengths of the diagonals. 8. Find the lengths of the diagonals of the kite and the rhombus below. m 3 m m 4 m 4 m 4 m m 3 m lengths of the diagonals of the kite: lengths of the diagonals of the rhombus: 6 m and 4 m 8 m and 6 m Problem 3 Finding the Area of a Kite Got It? What is the area of a kite with diagonals that are in. and 9 in. long? 9. Error Analysis Below is one student s solution. What error did the student make? A = ( + 9) Answers may vary. Sample: The student = () added the lengths of the diagonals instead =.5 of multiplying them.. Find the area of the kite. A 5 ()(9) 5 (8) The area of the kite is 54 in.. hapter 56

8 Problem 4 Finding the Area of a Rhombus Got It? A rhombus has sides cm long. If the longer diagonal is 6 cm, what is the area of the rhombus? Underline the correct words to complete the sentence.. The diagonals of a rhombus bisect each other / side and cm x cm cm are parallel / perpendicular. 3. Label the rhombus at the right. 4. The shorter diagonal is x x, or x. 8 cm cm cm 8 cm 5. Use the Pythagorean Theorem to 6. Find the area of the rhombus. find the value of x. x 8 5 x 64 5 x 5 36 x 5 6 A 5 ()(6) 5 (9) The area of the rhombus is 96 cm. Lesson heck Do you UNDERSTAND? Reasoning Do you need to know the lengths of the sides to find the area of a kite? Explain. 8. ross out the length you do NOT need to find the area of each triangle in a kite. each leg hypotenuse 9. Now answer the question. Explanations may vary. Sample: No. Each side of a kite is a hypotenuse of a right triangle. You need only the lengths of the legs to find the areas of the four right triangles that form the kite. Math Success heck off the vocabulary words that you understand. kite height of a trapezoid Rate how well you can find the area of a trapezoid, rhombus, or kite. Need to review Now I get it! 57 Lesson -

9 -3 Areas of Regular Polygons Vocabulary Review Write T for true or F for false. T. In a regular polygon, all sides are congruent. F. In a regular polygon, all angles are acute. 3. ross out the figure that is NOT a regular polygon Vocabulary Builder apothem (noun) AP uh them Related Words: center, regular polygon Definition: The apothem is the perpendicular distance from the center of a regular polygon to one of its sides. Use Your Vocabulary 4. Underline the correct word to complete the statement. In a regular polygon, the apothem is the perpendicular distance from the center to a(n) angle / side. 5. Label the regular polygon below using apothem, center, or side. center apothem side hapter 58

10 Problem Finding Angle Measures Got It? At the right, a portion of a regular octagon has radii and an apothem drawn. What is the measure of each numbered angle? 6. A regular octagon has 8 sides. 7. ircle the type of triangles formed by the radii of the regular octagon. equilateral isosceles right 8. Use the justifications below to find the measure of each numbered angle. m/ Divide 36 by the number of sides. 8 m/ 5 (m/) The apothem bisects the vertex angle of the 5 ( ) 5 triangle formed by the radii m/ m/3 5 8 Triangle Angle-Sum Theorem 9.5 m/3 5 8 Substitute. 3.5 m/3 5 8 Simplify. m/ Subtraction Property of Equality 9. Write the measure of each numbered angle. m/ 5 45 m/ 5.5 m/ Postulate - and Theorem -6 Postulate - If two figures are congruent, then their areas are equal. The isosceles triangles in the regular hexagon at the right are congruent. omplete each statement.. If the area of naob is 4 in., then the area of nbo is 4 in... If the area of nbo is 8 cm, then the area of nao is 6 cm. Theorem -6 Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter. omplete. A 5 ap. apothem: perimeter: 8 area: ()? 3. apothem: 5 perimeter: 3!3 area:? 5? 3!3 4. apothem: 5!3 perimeter: 6 area:? 5!3? 8 a p A 6 B O 59 Lesson -3

11 Problem Finding the Area of a Regular Polygon Got It? What is the area of a regular pentagon with an 8-cm apothem and.6-cm sides? 5. Label the regular pentagon with the lengths of the apothem and the sides. 6. Use the justifications below to find the perimeter. p 5 ns Use the formula for the perimeter of an n-gon. 5 5 (.6) Substitute for n and for s..6 cm 8 cm 5 58 Simplify. 7. Use the justifications below to find the area. A 5 ap Use the formula for the area of a regular polygon. 5? 8? 58 Substitute for a and for p. 5 3 Simplify. 8. The regular pentagon has an area of 3 cm. Problem 3 Using Special Triangles to Find Area Got It? The side of a regular hexagon is 6 ft. What is the area of the hexagon? Round your answer to the nearest square foot. 9. Use the information in the problem to complete the problem-solving model below. Know Need Plan I know that the length of each side of the regular The length of the apothem and the Draw a diagram to help find the length of the hexagon is 6 ft. perimeter apothem. Then use the perimeter and area formulas. Use the diagram at the right.. Label the diagram.. ircle the relationship you can use to find the length of the apothem. hypotenuse 5? shorter leg. omplete. length of shorter leg 5 length of longer leg (apothem) 5 ft 8!3 ft longer leg 5!3? shorter leg 3. Use the formula p 5 ns to find the perimeter of the hexagon. p 5 ns 5 6(6) ft 6 a 6 ft hapter 6

12 4. Now use the perimeter and the formula A 5 ap to find the area of the hexagon. A 5 ap 5 (8!3)(96) To the nearest square foot, the area of the hexagon is 665 ft. Lesson heck Do you UNDERSTAND? What is the relationship between the side length and the apothem in each figure? square regular hexagon equilateral triangle 45 s a 3 s a 3 s a 6. The radius and apothem form what type of triangle in each figure? square regular hexagon equilateral triangle triangle triangle triangle 7. omplete to show the relationship between the side length and the apothem. square regular hexagon equilateral triangle leg 5 leg longer leg 5!3? shorter leg longer leg 5!3? shorter leg a 5 s a 5!3? s s 5!3? a!3 a 5 s s 5!3 a Math Success heck off the vocabulary words that you understand. radius of a regular polygon Rate how well you can find the area of a regular polygon. Need to review apothem Now I get it! 6 Lesson -3

13 -4 Perimeters and Areas of Similar Figures Vocabulary Review. What does it mean when two figures are similar? Answers may vary. Sample: The figures have the same shape.. Are the corresponding angles of similar figures always congruent? Yes / No 3. Are the corresponding sides of similar figures always proportional? Yes / No 4. ircle the pairs of similar figures Vocabulary Builder radius (noun) RAY dee us (plural radii) Related Words: apothem, center Definition: The radius of a regular polygon is the distance from the center to a vertex. Main Idea: The radii of a regular polygon divide the polygon into congruent triangles. Use Your Vocabulary 5. ross out the segment that is NOT a radius of regular pentagon ABDE. OA OD OB OE O OF Underline the correct word(s) to complete each sentence. 6. The radii of a regular polygon are / are not congruent. 7. The triangles formed by the radii and sides of regular pentagon ABDE B A F O E D are / are not congruent. hapter 6

14 Theorem -7 Perimeters and Areas of Similar Figures If the scale factor of two similar figures is a b, then () the ratio of their perimeters is a b and () the ratio of their areas is a b. 8. The name for the ratio of the length of one side of a figure to the length of the corresponding side of a similar figure is the 9. scale factor 9. If the scale factor of two figures is, then the ratio of their perimeters is.. If the scale factor of two figures is 3 3 x, then the ratio of their perimeters is. x. If the scale factor of two figures is 3 3 5, then the ratio of their areas is. 5. If the scale factor of two figures is x then the ratio of their areas is. x Problem Finding Ratios in Similar Figures Got It? Two similar polygons have corresponding sides in the ratio 5 : 7. What is the ratio (larger to smaller) of their perimeters? What is the ratio (larger to smaller) of their areas? 3. ircle the similar polygons that have corresponding sides in the ratio 5 : 7. 5 Underline the correct word to complete each sentence. 4. In similar figures, the ratio of the areas / perimeters equals the ratio of corresponding sides. 5. In similar figures, the ratio of the areas / perimeters equals the ratio of the squares of corresponding sides. 6. omplete. 4 ratio (larger to smaller) ratio (larger to smaller) ratio (larger to smaller) of corresponding sides of perimeters of areas Lesson -4

15 Problem Finding Areas Using Similar Figures Got It? The scale factor of two similar parallelograms is 3 4. The area of the larger parallelogram is 96 in..what is the area of the smaller parallelogram? Write T for true or F for false. F 7. The ratio of the areas is 3 4. T 8. The ratio of the areas is Use the justifications below to find the area A of the smaller parallelogram. 9 5 A 6 96 Write a proportion. 6A 5 (96) 9 ross Products Property 6A Multiply. 6A Divide each side by A 5 54 Simplify.. The area of the smaller parallelogram is 54 in.. Problem 3 Applying Area Ratios Got It? The scale factor of the dimensions of two similar pieces of window glass is 3 : 5. The smaller piece costs $.5. How much should the larger piece cost?. Use the information in the problem to complete the reasoning model below. Think The ratio of areas is the square of the scale factor. I can use a proportion to find the cost c of the larger piece to the nearest hundredth. Write Ratio of areas 3 : c 9 : 5 9 c c c 9 c The larger piece of glass should cost about $ hapter 64

16 Problem 4 Finding Perimeter Ratios Got It? The areas of two similar rectangles are 875 ft and 35 ft. What is the ratio of their perimeters? 3. The scale factor is found below. Use one of the reasons listed in the blue box to justify each step. a b 5 35 Write a proportion. 875 a b a b 5 3 5!5 a b 5 3 5!5?!5!5 a b 5 3!5 5 Simplify. Take the positive square root of each side. Rationalize the denominator. Simplify. Rationalize the denominator. Simplify. Simplify. Take the positive square root of each side. Write a proportion. 4. The ratio of the perimeters equals the scale factor 3!5 : 5. Lesson heck Do you UNDERSTAND? Reasoning The area of one rectangle is twice the area of another. What is the ratio of their perimeters? How do you know? 5. Let x and y be the sides of the smaller rectangle. omplete. area of smaller rectangle area of larger rectangle ratio of larger to smaller areas Math Success heck off the vocabulary words that you understand. similar polygons radius perimeter area Rate how well you can find the perimeters and areas of similar polygons. Need to review xy xy xy : 6. Find the square root of the ratio of larger to smaller areas to find the scale factor.!xy!xy 5 xy Å xy 5 Å 5! 7. The ratio of perimeters is! : because the scale factor is! : Now I get it! xy 65 Lesson -4

17 -5 Trigonometry and Area Vocabulary Review. Underline the correct word to complete the sentence. Area is the number of cubic / square units needed to cover a given surface.. ircle the formula for the area of a triangle. A 5 bh A 5 bh A 5 h(b b ) A 5 d d Vocabulary Builder trigonometry (noun) trig uh NAHM uh tree Other Word Form: trigonometric (adjective) Related Words: cosine, sine, tangent Definition: Trigonometry is the study of the relationships among two sides and an angle in a right triangle. Main Idea: You can use trigonometry to find the area of a regular polygon. Use Your Vocabulary omplete each sentence with the word trigonometry or trigonometric. 3. The sine, cosine, and tangent ratios are 9 ratios. 4. This year I am studying 9 in math. Draw a line from each trigonometric ratio in olumn A to its name in olumn B. olumn A length of opposite leg 5. length of hypotenuse length of adjacent leg length of hypotenuse length of opposite leg length of adjacent leg olumn B cosine sine tangent trigonometric trigonometry hapter 66

18 Problem Finding Area Got It? What is the area of a regular pentagon with 4-in. sides? Round your answer to the nearest square inch. 8. Underline the correct words to complete the sentence. To find the area using the formula A 5 ap, you need to know the length of the apothem / radius and the perimeter / width of the pentagon. 9. In the regular pentagon at the right, label center, apothem R, and radii D and E.. The perimeter of the pentagon is 5? 4 in., or in. 4 in.. The measure of central angle DE is 36, or 7. 5 omplete Exercises and 3.. m/dr 5 m/de 3. DR 5 DE 5? 7 5? 4 D R E Use your results from Exercises and 3 to label the diagram below. a 36 D R in. 5. ircle the equation you can use to find the apothem a. tan a tan 368 5a a tan a tan a tan 78 5a 6. Use the justifications below to find the apothem and the area. tan 78 5 Use the tangent ratio. a a? tan Multiply each side by a. a 5 tan 368 Divide each side by tan 368. A 5 ap Write the formula for the area of a regular polygon. 5?? Substitute for a and p. tan 368 < Use a calculator. 7. To the nearest square inch, the area of the regular pentagon is 8 in.. 67 Lesson -5

19 Problem Finding Area Got It? A tabletop has the shape of a regular decagon with a radius of 9.5 in. What is the area of the tabletop to the nearest square inch? 8. omplete the problem-solving model below. Know The radius and number of sides of the decagon Need The apothem and the length of a side Plan Use trigonometric ratios to find the apothem and the length of a side. 9. Look at the decagon at the right. Explain why the measure of each central angle of a decagon is 36 and m/ is 8. There are central angles and The apothem a bisects a central angle, so ml 5 (36), or a. Use the cosine ratio to find the apothem a.. Use the sine ratio to find x. cos 88 5 a sin 88 5 x ? cos 88 5a 9.5? sin 88 5x. Use the justifications below to find the perimeter. x p 5? length of one side perimeter 5 number of sides times length of one side 5?? x The length of each side is x. 5?? 9.5(sin 88) Substitute for x. 5 9? sin 88 Simplify. 3. Find the area. Use a calculator. A 5 ap 5? 9.5(cos 88)? 9(sin 88) N To the nearest square inch, the area of the tabletop is 65 in.. Theorem -8 Area of a Triangle Given SAS The area of a triangle is half the product of the lengths of two sides and the sine of the included angle. 5. omplete the formula below. c B a Area of n AB 5 bc(sin A ) A b hapter 68

20 Problem 3 Finding Area Got It? What is the area of the triangle? Round your answer to the nearest square inch. 6. omplete the reasoning model below. 6 in. in. 34 Think I know the lengths of two sides and the measure of the included angle. Write Side lengths: in. and 6 in. Angle measure: 34 I can use the formula for the area of a triangle given SAS. A 6 sin To the nearest square inch, the area of the triangle is 45 in.. Lesson heck Do you UNDERSTAND? Error Analysis Your classmate needs to find the area of a regular pentagon with 8-cm sides. To find the apothem, he sets up and solves a trigonometric ratio. What error did he make? Explain. 8. The lengths of the legs of the triangle in the regular pentagon are a and 4 cm. length of opposite leg 9. The tangent of the 368 angle is length of adjacent leg, or 4. a 3. Explain the error your classmate made. He used a 4 instead of 4 a for the tangent ratio. Math Success heck off the vocabulary words that you understand. area Rate how well you can use trigonometry to find area. Need to review trigonometry Now I get it! 36 a 8 cm a tan 36 4 a 4 tan Lesson -5

21 -6 ircles and Arcs Vocabulary Review. Is a circle a two-dimensional figure? Yes / No. Is a circle a polygon? Yes / No 3. Is every point on a circle the same distance from the center? Yes / No 4. ircle the figure that is a circle. Vocabulary Builder arc (noun) ahrk Major arc AB Minor arc A Definition: An arc is part of a circle. Related Words: minor arc, major arc, semicircle Example: Semicircle AB is an arc of the circle. Use Your Vocabulary Underline the correct word to complete each sentence. 5. A minor arc is larger / smaller than a semicircle. 6. A major arc is larger / smaller than a semicircle. 7. You use two / three points to name a major arc. 8. You use two / three points to name a minor arc. 9. ircle the name of the red arc. JK KL. ircle the name of the blue arc. LJK LKJ B O A J O JK KL LJK LKJ L K hapter 7

22 Problem Naming Arcs Got It? What are the minor arcs of (A? S P Draw a line from each central angle in olumn A to its corresponding minor arc in olumn B. A Q olumn A olumn B R. /PAQ RS. /QAR SP 3. /RAS PQ 4. /SAP QR 5. /SAQ SQ 6. The minor arcs of (A are PQ, QR, RS, SP, and SQ. Key oncepts Arc Measure and Postulate - Arc Measure The measure of a minor arc is equal to the measure of its corresponding central angle. The measure of a major arc is the measure of the related minor arc subtracted from 36. The measure of a semicircle is 8. Use (S at the right for Exercises 7 and 8. R 7. m RT 5 m/rst m TQR S m RT T Q Postulate - Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. m AB 5 m AB m B B Use the circle at the right for Exercises 9 and. 9. If m AB 5 4 and m B 5, then m AB 5 4. A. If m AB 5 x and m B 5 y, then m AB 5 x y. Problem Finding the Measures of Arcs Got It? What are the measures of PR, RS, PRQ, and PQR in (? omplete.. m/pr 5 77, so m PR P 77 R S Q 8 7 Lesson -6

23 . m/rs 5 m/ps m/pr m/rs 5 3, so m RS mprq 5 m PR m RS m SQ P 77 Q R S mpqr 5 36 m PR Theorem -9 ircumference of a ircle The circumference of a circle is p times the diameter. 5 pd or 5 pr 6. Explain why you can use either 5 pd or 5 pr to find the circumference of a circle. r d O Explanations may vary. Sample: The length of the diameter is twice the length of the radius. Problem 3 Finding a Distance Got It? A car has a circular turning radius of 6. ft. The distance between the two front tires is 4.7 ft. How much farther does a tire on the outside of the turn travel than a tire on the inside? 7. The two circles have the same center. To find the radius of the inner circle, do you add or subtract? omplete. 8. radius of outer circle 5 6. radius of inner circle circumference of outer circle 5 pr 5 p? ? p circumference of inner circle 5 pr 5 p?.4 5.8? p 3. Find the differences in the two distances traveled. Use a calculator. 3.? p.8? p 5 9.4? p < subtract 6. ft 4.7 ft 3. A tire on the outer circle travels about 3 ft farther. hapter 7

24 Theorem - Arc Length The length of an arc of a circle is the product of the ratio the circumference of the circle. 3. omplete the formula below. length of 5 m AB 36? πr 5 m AB AB 36? πd Write T for true or F for false. measure of the arc 36 and T T 33. The length of an arc is a fraction of the circumference of a circle. 34. In (O, m AB 5 m/aob. Lesson heck Do you UNDERSTAND? Error Analysis Your class must find the length of AB. A classmate submits the following solution. What is the error? mab Length of AB = r 36 = (4) 36 = 9 m A O 4 m B Is A a semicircle? Yes / No 36. Does m AB ? Yes / No 37. Is the length of the radius 4? Yes / No 38. What is the error? Answers may vary. Sample: The student substituted the diameter instead of the radius for r. Math Success heck off the vocabulary words that you understand. circle minor arc major arc circumference Rate how well you can use central angles, arcs, and circumference. Need to review Now I get it! 73 Lesson -6

25 -7 Areas of ircles and Sectors Vocabulary Review. Explain how the area of a figure is different from the perimeter of the figure. Area is the measure of the space inside a figure while perimeter is the distance around the figure.. ircle the formula for the area of a parallelogram. A 5 bh A 5 bh A 5 h(b b ) A 5 d d 3. Find the area of each figure. 3 m 6 cm 6 ft 5 m 9 cm ft A 5 5 m A 5 7 cm A 5 6 ft Vocabulary Builder sector (noun) SEK tur Definition: A sector of a circle is a region bounded by an arc of the circle and the two radii to the arc s endpoints. Main Idea: The area of a sector is a fractional part of the area of a circle. Use Your Vocabulary 4. Name the arc and the radii that are the boundaries of the shaded sector. arc AB radii A and B 5. ircle the name of the shaded sector. sector RST R T sector AB sector AB sector BA 6. The shaded sector is what fractional part of the area of the circle? Explain. 4 ; AB is a 98 arc and S A B hapter 74

26 Theorem - Area of a ircle The area of a circle is the product of p and the square of the radius. A 5 pr omplete each statement. 7. If the radius is 5 ft, then A 5 p? 5? If the diameter is.8 cm, then A 5 p?.9?.9. O r Problem Finding the Area of a ircle Got It? What is the area of a circular wrestling region with a 4-ft diameter? 9. The radius of the wrestling region is ft.. omplete the reasoning model below. 4 ft Think I can use the formula for the area of a circle. I can subtitute the radius into the formula and then simplify. Write A πr π 44 π I can use a calculator to find the approximate area The area of the wrestling region is about 385 ft. Theorem - Area of a Sector of a ircle The area of a sector of a circle is the product of the ratio and the area of the circle. Area of sector AOB 5 m AB 36? pr omplete measure of the arc measure of the arc 36 measure of the arc 36 area of the sector ? π? r ? π? r 3 3 A r O r B 75 Lesson -7

27 Problem Finding the Area of a Sector of a ircle Got It? A circle has a radius of 4 in. What is the area of a sector bounded by a 458 minor arc? Leave your answer in terms of π. 4. At the right is one student s solution. What error did the student make? The student did not square the radius. 5. Find the area of the sector correctly. area of sector = 45 π(4) 36 = π(4) 8 = π area of sector ? π(4) 5 8? π(6) 5 π 6. The area of the sector is π in.. Key oncept Area of a Segment The area of a segment is the difference of the area of the sector and the area of the triangle formed by the radii and the segment joining the endpoints. Area of sector Area of triangle Area of segment Problem 3 Finding the Area of a Segment of a ircle Got It? What is the area of the shaded segment shown at the right? Round your answer to the nearest tenth. 7. Use the justifications below to find the area of sector PQR. area of sector PQR 5 m PR? pr Use the formula for the area of a sector ? p( 4 ) Substitute. 4 m Q P R 5 4? p Simplify. 8. npqr is a right triangle, so the base is 4 m and the height is 4 m. hapter 76

28 9. Find the area of npqr. A 5 bh 5 (4)(4) 5 8. omplete to find the area of the shaded segment. Use a calculator. area of shaded segment 5 area of sector PQR area of npqr 5 4? p 8 < To the nearest tenth, the area of the shaded segment is 4.6 m. Lesson heck Do you UNDERSTAND? Reasoning Suppose a sector in (P has the same area as a sector in (O. an you conclude that (P and (O have the same area? Explain. Use the figures at the right for Exercises 4.. Find the area of sector AO in (O. 3. Find the area of sector RPT in (P. area of sector ? π(8) area of sector ? π(4) O 45 8 m A 4. Do the sectors have the same area? an you conclude that the circles have the same area? Explain. Explanations may vary. Sample: Yes; no; the circles do not have the same area because their radii are different lengths. Math Success heck off the vocabulary words that you understand. sector of a circle segment of a circle area of a circle Rate how well you can find areas of circles, sectors, and segments. Need to review 5 8? π(64) 5 8π Now I get it! 5? π(6) 5 8π T 8 m P R 77 Lesson -7

29 -8 Geometric Probability Vocabulary Review Write T for true or F for false. T F. A point indicates a location and has no size.. A line contains a finite number of points. 3. Use the diagram at the right. ircle the segment that includes point S. PR PT QR P Q R S T Vocabulary Builder probability (noun) prah buh BIL uh tee Related Term: geometric probability theoretical probability number of favorable outcomes P(event) number of possible outcomes Definition: The probability of an event is the likelihood l that the event will occur. Main Idea: In geometric probability, favorable outcomes and possible outcomes are geometric measures such as lengths of segments or areas of regions. Use Your Vocabulary 4. Underline the correct words to complete the sentence. The probability of an event is the ratio of the number of favorable / possible outcomes to the number of favorable / possible outcomes. 5. There are 7 red marbles and 3 green marbles in a bag. One marble is chosen at random. Write the probability that a green marble is chosen. Write as a fraction. 3 P(green) Write as a decimal. Write as a percent..3 3 % hapter 78

30 Key oncept Probability and Length or Area Probability and Length Point S on AD is chosen at random. The probability that S is on B is the ratio of the length of B to the length of AD. A B D P(S on B) 5 B AD omplete. A 6. P(S on A ) 5 7. P(S on AB) 5 AD Probability and Area Point S in region R is chosen at random. The probability that S is in region N is the ratio of the area of region N to the area of region R. P(S in region N) 5 area of region N area of region R AB AD R N 8. Find the probability for the given areas. area of region R 5 cm area of region N 5 3 cm P(S in N) 5 3 Problem Using Segments to Find Probability Got It? Point H on ST is selected at random. What is the probability that H lies on SR? 9. Find the length of each segment. Problem S Q R T length of SR 5 u u 5 length of ST 5 u u 5. Find the probability. 8 6 length of SR 6 P(H on SR) length of ST. The probability that H is on SR is, or 5 %. Using Segments to Find Probability Got It? Transportation A commuter train runs every 5 min. If a commuter arrives at the station at a random time, what is the probability that the commuter will have to wait no more than 5 min for the train?. ircle the time t (in minutes) before the train arrives that the commuter will need to arrive in order to wait no more than 5 minutes. # t # 5 5, t #, t # 5 5, t #, t # Lesson -8

31 3. ircle the diagram that models the situation. A B A B A B omplete. length of favorable segment 5 5 length of entire segment Find the probability. P(waiting no more than 5 min) 5 length of favorable segment length of entire segment 5 5, or The probability of waiting no more than 5 min for the train is 5, or %. Problem 3 Using Area to Find Probability Got It? A triangle is inscribed in a square. Point T in the square is selected at random. What is the probability that T lies in the shaded region? 5 in. 7. omplete the model below to write an equation. Define Let s the area of the shaded region. area of the area of the Relate is minus shaded region square Write s 5 area of the triangle 8. Now solve the equation to find the area of the shaded region. s 5 5? 5? 5 5 5? Find the probability. P(point T is in shaded region) 5 area of shaded region area of square.5 5, or The probability that T lies in the shaded region is, or 5 %. hapter 8

32 Problem 4 Using Area to Find Probability Got It? Archery An archery target has 5 colored scoring zones formed by concentric circles. The target s diameter is cm. The radius of the yellow zone is. cm. The width of each of the other zones is also. cm. If an arrow hits the target at a random point, what is the probability that it hits the yellow zone?. The radius of the target is, or 6 cm.. Find the probability. Write the probability as a decimal. P(arrow hits yellow zone) 5 area of yellow zone area of entire target 5 p(.) p( 6 ) Explain why the calculation with p is not an estimate. Answers may vary. Sample: I divide π out before finding the quotient. 4. The probability that the arrow hits the yellow zone is.4, or 4 %. Lesson heck Do you UNDERSTAND? Reasoning In the figure at the right, SQ QT 5. What is the probability that a point on ST chosen at random will lie on QT? Explain. 5. If SQ 5 x, then QT 5 x and ST 5 3x. 6. What is P(point on QT )? Explain. 3 ; length of QT 5 x length of ST 3x 5 3 Math Success heck off the vocabulary words that you understand. Need to review Now I get it! S Q T length area geometric probability Rate how well you can use geometric probability. 8 Lesson -8