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1. in the Figure, Heptagon ABCDEFG Is Inscribed in ⊙P. Identify the Center

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10-4 Areas of Regular Polygons and Composite Figures 1. In the figure, heptagon ABCDEFG is inscribed in Find the area of each regular polygon. Round to ⊙P. Identify the center, a radius, an apothem, and a the nearest tenth. central angle of the polygon. Then find the measure of a central angle. 2. SOLUTION: An equilateral triangle has three congruent sides. Draw an altitude and use the Pythagorean Theorem to find the height. SOLUTION: center: point P, radius: , apothem: , central angle: . A regular heptagon has 7 congruent sides and angles. Thus, the measure of each central angle of heptagon ABCDEFG is . ANSWER: center: point P, radius: , apothem: , central angle: Find the area of the triangle. ANSWER: eSolutions Manual - Powered by Cognero Page 1 10-4 Areas of Regular Polygons and Composite Figures 3. 4. SOLUTION: SOLUTION: The polygon is a square. Form a right triangle. First, find the apothem of the polygon. The central angle of a regular hexagon is Half of the central angle is 30 degrees. This makes this triangle a 30°-60°-90 special right triangle. Use the Pythagorean Theorem to find x. Find the area of the square. Use the formula for the Area of a Regular Polygon; ANSWER: ANSWER: 166.3 cm² eSolutions Manual - Powered by Cognero Page 2 10-4 Areas of Regular Polygons and Composite Figures 5. POOLS Kenton’s job is to cover the community pool C 75 in² during fall and winter. Since the pool is in the shape D in² of an octagon, he needs to find the area in order to SOLUTION: have a custom cover made. If the pool has the To determine the area of the composite shape made dimensions shown at the right, what is the area of the up of 6 equilateral triangles and one regular hexagon, pool? start by finding the area of the individual shapes. The area of one equilateral triangle with a side length of 5 in. can be found by using 30°-60°-90° special right triangle knowledge: SOLUTION: Since the polygon has 8 sides, the polygon can be Similarly, since the hexagon is composed on 6 divided into 8 congruent isosceles triangles, each with equilateral triangles, the apothem of the regular a base of 5 ft and a height of 6 ft. hexagon is the same as the height of the equilateral Find the area of one triangle. triangle: Since there are 8 triangles, the area of the pool is 15 · 8 or 120 square feet. Now, combine the different shapes to get the entire ANSWER: area: The correct choice is D. 6. MULTIPLE CHOICE The figure shown is composed of a regular hexagon and equilateral ANSWER: triangles. Which of the following best represents the D area? 7. BASKETBALL The basketball court in Jeff’s school is painted as shown. A 37.5 in² Note: Art not drawn to scale. B in² a. What area of the court is blue? Round to the eSolutions Manual - Powered by Cognero Page 3 10-4 Areas of Regular Polygons and Composite Figures nearest square foot. SENSE-MAKING In each figure, a regular b. What area of the court is red? Round to the polygon is inscribed in a circle. Identify the nearest square foot. center, a radius, an apothem, and a central angle SOLUTION: of each polygon. Then find the measure of a central angle. a. The small blue circle in the middle of the floor has a diameter of 6 feet so its radius is 3 feet. The blue sections on each end are the area of a rectangle minus the area of half the red circle. The rectangle has dimensions of 12 ft by 19 ft. The diameter of the red circle is 12 feet so its radius is 6 feet. 8. Area of blue sections = Area of small blue circle + 2 SOLUTION: [Area of rectangle – Area of red circle ÷ 2] Center: point X, radius: , apothem: , central angle: , A square is a regular polygon with 4 sides. Thus, the measure of each central angle of square RSTVW is or 72. ANSWER: center: point X, radius: , apothem: , central 2 So, the area of the court that is blue is about 371 ft . angle: VXT, 72° b. The two red circles on either end of the court each have a diameter of 12 feet or a radius of 6 feet. The large circle at the center of the court has a diameter of 12 feet so it has a radius of 6 feet. The inner blue circle has a diameter of 6 feet so it has a radius of 3 feet. 9. SOLUTION: Area of red sections = 2 [Area of end red circles] – Center: point R, radius: , apothem: , central [Area of large center circle – Area of blue center angle: . The measure of each central angle of circle] JKLMNOPQ is or 45°. ANSWER: center: point R, radius: , apothem: , central angle: KRL, 60° So, the area of the court that is red is about 311 ft2. ANSWER: a. b. eSolutions Manual - Powered by Cognero Page 4 10-4 Areas of Regular Polygons and Composite Figures Find the area of each regular polygon. Round to SOLUTION: the nearest tenth. The formula for the area of a regular polygon is , so we need to determine the perimeter and the length of the apothem of the figure. A regular pentagon has 5 congruent central angles, so the measure of central angle is or 72. 10. SOLUTION: An equilateral triangle has three congruent sides. Draw an altitude and use the Pythagorean Theorem to find the height. Apothem is the height of the isosceles triangle ABC. Triangles ACD and BCD are congruent, with ∠ACD = ∠BCD = 36. Use the Trigonometric ratios to find the side length and apothem of the polygon. Find the area of the triangle. ANSWER: Use the formula for the area of a regular polygon. ANSWER: 11. eSolutions Manual - Powered by Cognero Page 5 10-4 Areas of Regular Polygons and Composite Figures 12. 13. SOLUTION: SOLUTION: A regular hexagon has 6 congruent central angles A regular octagon has 8 congruent central angles, that are a part of 6 congruent triangles, so the from 8 congruent triangles, so the measure of central angle is 360 ÷ 8 = 45. measure of the central angle is = 60. Apothem is the height of the isosceles triangle Apothem is the height of equilateral triangle ABC and it splits the triangle into two congruent ABC and it splits the triangle into two 30-60-90 triangles. triangles. Use the Trigonometric ratio to find the side length of Use the trigonometric ratio to find the apothem of the the polygon. polygon. AB = 2(AD), so AB = 8 tan 30. ANSWER: ANSWER: 14. CARPETING Ignacio's family is getting new carpet in their family room, and they want to determine how much the project will cost. a. Use the floor plan shown to find the area to be carpeted. eSolutions Manual - Powered by Cognero Page 6 10-4 Areas of Regular Polygons and Composite Figures b. If the carpet costs $4.86 per square yard, how SENSE-MAKING Find the area of each figure. much will the project cost? Round to the nearest tenth if necessary. 15. SOLUTION: SOLUTION: a. The longer dotted red line divides the floor into two quadrilaterals. The quadrilateral formed on top will have four right angles, so it is a rectangle with a base of 24 feet. The height of the rectangle is 17 – 6 = 11 feet.The longer dotted red side and the bottom side (9 ft side) are both perpendicular to the shorter dotted red side (6 ft side) so they are parallel to each other. Since the quadrilateral on the bottom has two parallel sides, it is a trapezoid with a height of 6 feet and bases of length 9 feet and 24 feet (opposite sides of a rectangle are congruent). The area of the room will be the sum of the area of the rectangle and the area ANSWER: of the trapezoid. So, the area of the floor to be carpeted is 363 ft2. b. At $4.86 per yard, the project will cost: ANSWER: a. b. $196.02 eSolutions Manual - Powered by Cognero Page 7 10-4 Areas of Regular Polygons and Composite Figures 18. SOLUTION: 16. SOLUTION: ANSWER: 19. CRAFTS Latoya’s greeting card company is making envelopes for a card from the pattern shown. a. Find the perimeter and area of the pattern? Round to the nearest tenth. b. If Latoya orders sheets of paper that are 2 feet by 4 feet, how many envelopes can she make per sheet? ANSWER: 17. SOLUTION: SOLUTION: a. To find the perimeter of the envelope, first use the Pythagorean theorem to find the missing sides of the isosceles triangle on the left. ANSWER: The base of the isosceles triangle is 5.5 inches, so the height will bisect the base into two segments that eSolutions Manual - Powered by Cognero Page 8 10-4 Areas of Regular Polygons and Composite Figures each have a length of 2.75 inches. a width of 2 feet or 24 inches. 24 ÷ 5.75 ≈ 4.17 so 4 patterns can be placed widthwise on the paper. The maximum length of the pattern is inches. The sheet of paper has a length of 4 feet or 48 inches. so 4 patterns can be placed lengthwise on the paper. So, each side of the isosceles triangle is about 3.83 inches long. Four patterns across by four patterns high will make a total of 4 × 4 or 16. Find the sum of the lengths of all the sides of the So, Latoya can make 16 cards per sheet.
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