Applications for Triangles

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1 Not drawn to scale Applications for Triangles in. 40 in. 33 in in in in in in in in. 2 none of these Find the area of a parallelogram with the given vertices. 3. P(1, 3), Q(3, 3), R(7, 8), S(9, 8) 10 units 2 5 units 2 20 units 2 none of these 4. An isosceles triangle has area of 110 ft 2. If the base is 14 ft, what is the length of the legs? Round your answer to the nearest tenth. 21 ft 17.2 ft 14.8 ft ft 5. A fly lands at random at a point on the gri Find the probability of the fly landing on the figure. 6. Find the area of the rhombus. Leave your answer in simplest radical form

2 P 18 Q 7. In trapezoid PQRS,.. Find the area of PQRS. Leave your answer in simplified radical form. 8. Given the regular polygon, find the measure of each numbered angle S R Given the regular hexagon, find the measure of each numbered angle Given a regular hexagon, find the measures of the angles formed by (a) two consecutive radii and (b) a radius and a side of the polygon. 40 ; ; ; ; The area of a regular hexagon is 35 in. 2. Find the length of a side. Round your answer to the nearest tenth. 3.7 in. 4.8 in. 6.4 in in. 12. Find the area of a regular hexagon with side length of 8 m. Round your answer to the nearest tenth m m m m Find the area of an equilateral triangle with side Find the area of the regular polygon. Round your answer to the nearest tenth.

3 13.07 in. 10 in in in in in You are planning to use a ceramic tile design in your new bathroom. The tiles are blue and white equilateral triangles. You decide to arrange the blue tiles in a hexagonal shape as shown. If the side of each tile measures 7 centimeters, what will be the exact area of each hexagonal shape? 7 cm 73.5 cm 2 98 cm 2 21 cm cm Find the area of an equilateral triangle with radius 8 m. Leave your answer in simplest radical form. 96 m m 2 18 m 2 12 m A regular hexagon has a perimeter of 150 m. Find its are Leave your answer in simplest radical form m m m m 2 2 The figures are similar. The area of one figure is given. Find the area of the other figure to the nearest whole number. 18. The area of a regular octagon is 35 cm. What is the area of a regular octagon with sides three times as large? 315 cm 225 cm 175 cm 105 cm Find the area of the regular polygon. Give the answer to the nearest tenth. 19. pentagon with side 10 cm cm 20. hexagon with side 8 yd yd 34.4 cm 12 yd cm 41.6 yd 172 cm yd

4 21. decagon with side 4 cm cm cm cm cm 22. dodecagon with perimeter 108 cm cm cm cm cm 23. pentagon with radius 8 m m m 30.4 m m 24. hexagon with radius 5 in in in in in. 25. The Ruffs are planning to buy an above-ground swimming pool shaped as a regular octagon. The radius of the octagon is 9 feet. To the nearest tenth, find the area of the surface of the water in the pool ft ft 94.8 ft ft 26. Find the area of the triangle. Give the answer to the nearest tenth. The drawing may not be to scale. 15 cm??? 9 cm cm 63.4 cm 23.1 cm cm 4.7 m??? 6.1 m 10.5 m 9.8 m 19.6 m 21.0 m 28. A gardener needs to cultivate a triangular plot of lan One angle of the garden is 47, and two sides adjacent to the angle are 77 feet and 76 feet. To the nearest tenth, what is the area of the plot of land? ft ft ft ft 29. A park in a subdivision is triangular-shape Two adjacent sides of the park are 573 feet and 536 feet. The angle between the sides is 58. To the nearest unit, find the area of the park in square yards.

5 32,557 yd 14,470 yd 28,940 yd 43,410 yd 30. Divers looking for a sunken ship have defined the search area as a triangle with adjacent sides of length 2.75 miles and 1.32 miles. The angle between the sides of the triangle is 35. To the nearest hundredth, find the search are 2.08 mi 2.97 mi 1.04 mi 1.49 mi 31. Two triangles each have adjacent sides of length 120 feet and 180 feet. The first triangle has an angle between the two sides of 40, while the second triangle has an angle between the two sides of 60. What is the approximate difference between the areas of the two triangles? 4822 ft 2873 ft 5746 ft 2411 ft

6 32. Grade 7 students were surveyed to determine how many hours a day they spent on various activities. The results are shown in the circle graph below. Find the measure of each central angle in the circle graph. Sleeping Eating ; ; ; ; Name the minor arc and find its measure. A D C ) 115 B arc ADB; 30 arc AB; 115 arc ADB; 245 arc AB; Name the major arc and find its measure. A D C )50 B

7 arc ADB; 50 arc AB; 50 arc ADB; 310 arc AB; Find the area of the shaded portion of the figure. Each vertex of square ABCD is at the center of a circle. Leave your answer in terms of. 36. A team in science class placed a chalk mark on the side of a wheel and rolled the wheel in a straight line until the chalk mark returned to the same position. The team then measured the distance the wheel had rolled and found it to be 35 cm. To the nearest tenth, what is the area of the wheel? cm cm cm cm 2 Find the area of the circle. Leave your answer in terms of. 37. The figure represents the overhead view of a deck surrounding a hot tu What is the area of the deck? Round to the nearest tenth m m m m Find the area of the shaded portion of the figure. Dimensions are in feet. Leave your answer in terms of π.

8 none of these 39. Find the area of the figure to the nearest tenth in in in in A 4-ft-by-5-ft dock is anchored in the middle of a lake. The bow of a boat is tied to a corner of the dock with a 5-ft rope as shown in the picture. Find the area of the region in which the bow of the boat can travel. Round your answer to the nearest foot. Not drawn to scale 60 ft 2 58 ft 2 81 ft 2 none of these 41. Find the exact area of the shaded region.

9 none of these 42. Find the area of the shaded region. Leave your answer in terms of and in simplest radical form. none of these 43. What is the probability that a point chosen at random on the grid will lie in the unshaded region?

10 44. The radius of the bull s-eye of the dartboard is 8 inches. The radius of each concentric circle is 8 inches more than the radius of the circle inside it. If a dart lands at random on the dartboard, what is the probability that the dart will hit in area C? A B C D 45. Find the probability that a point chosen at random will lie in the shaded are

11 46. A circular dartboard has a radius of 2 meters and a red circle in the center. Assume you hit the target at a random point. For what radius of the red center region does P(hitting red) = 0.6? 77 m 1.2 m 1.55 cm 1.32 m 47. The regular polygon has radius 9 m. Find each angle measure to the nearest tenth of a degree, each linear measure to the nearest tenth of a meter, and the square measure to the nearest square meter. G H F E O A X B D C OX AB e. the perimeter f. the area Essay 48. Find the area of a regular hexagon with sides 2 cm long. Leave your answer in simplest radical form. Use your answer from part (a) to find the area of a regular hexagon of side length An outdoor deck for a new restaurant forms a square with radius 30 feet. Draw and label a diagram of the deck. Explain your diagram. Find the perimeter of the deck. Explain your method for finding the perimeter. Find the area of the deck. Explain your method for finding the are Show a method for finding the area of the deck that is different from the method you used in part (c). 50. Use this triangle. 7 in.??????

12 Find the length of the hypotenuse using the special relationships for a triangle. Explain the relationships. Write the exact value for sin 45 as a fraction in simplest form. Use the formula for the area of a triangle given SAS to find the area of the triangle. Explain your steps. Use a different method to find the area of the triangle. Explain your steps. 51. The diameter of a basketball rim is 18 inches. A standard basketball has a circumference of 30 inches. About how much room is there between the ball and the rim in a shot in which the ball goes in exactly in the center of the rim? Show your work. 52. Jason designed an arch made of wrought iron for the top of a mall entrance. The 11 segments between the two concentric circles are each 1.25 m long. Find the total length of wrought iron used to make the structure. Round the answer to the nearest meter.

13 Applications Answer Section MULTIPLE CHOICE 1) A 2) B 3) A 4) B 5) B 6) D 7) A 8)B 9) C 10) B 11) A 12) B 13) A 14) D 15) A 16) B 17) B 18 A 19) D 20)D 21) A 22) B 23) B 24) C 25) D 26) B 27) A 28) B 29) B 30) C 31) D 32 A 33) B 34 C 35 A 36)B 37) B 38 C 39) A 40 A 41)B 42)C 43)A 44 B 45 A 46) C 47. ANS: m 6.8 m e m f. 229 m 48. ANS: [4] Answers may vary. Sample: The central angle of one of the triangles in the hexagon is. The altitude of the triangle is because it is a - - right triangle.

14 ??? 2 cm??? 1 cm The area of the hexagon is therefore. Because regular hexagons are similar, their areas will be proportional to the square of their similarity ratio.. cm ANS: [4] 30 ft a x The radius of the square from its center to a corner is 30 feet. A triangle can be formed where x is half the side length and a is the apothem. The triangle in the diagram is a triangle. The hypotenuse 30 is x and a and a = x. To find the value of x, or a, write and solve an equation. 30 = x Write an equation. = Divide each side by x Use a calculator. So, x = a 21.2 and the perimeter is about 2(21.2)(4) feet, or feet. Now, use the formula for the area of a regular polygon. A =

15 = Substitute. = Simplify. The area of the deck is about 1798 square feet. Another way to find the area is to use the formula for the area of a square. The length of the side of the deck is 2(21.2), or The area of the deck is about 1798 square feet. 50. ANS: [4] In a triangle, the length of the hypotenuse is the length of the short leg times, so the length of the hypotenuse is 7 in. The exact value of sin 45 is = =. Area of a triangle = = b = 7, c = 7, A = 45 = Simplify. = 24.5 The area of the triangle is 24.5 in.. Another formula for the area of a triangle if the base length and height are known is. For this triangle, b = 7 and h = 7 since they are the legs of a right triangle. A = = b = 7, h = 7 = Simplify. = 24.5 The area of the triangle is 24.5 in ANS

16 [4] Answers may vary. Sample: If a basketball has a circumference of 30, then its diameter is. If the ball goes in exactly in the center of the rim, then there is a total of ( ) inches or inches on both sides of the ball. Therefore, there is half of this distance, or about 4.2 inches, between the ball and the rim. [3] correct methods used, but with a computational error [2] error in method [1] correct answer with no work shown 52. ANS: [4] Answers may vary. Sample: The total length of wrought iron used is the sum of the outer circumference plus the inner circumference plus the eleven segments. The length of wrought iron used is approximately 59 meters.

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