The scale factor between the blue diamond and the green diamond is, so the ratio of their areas is.
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1 For each pair of similar figures, find the area of the green figure. 1. The scale factor between the blue diamond and the green diamond is, so the ratio of their areas is. The area of the green diamond is 9 yd 2. For each pair of similar figures, use the given areas to find the scale factor from the blue to the green figure. Then find x. 3. The scale factor between the blue triangle and the green triangle is, so the ratio of their areas is. The scale factor is or. esolutions Manual - Powered by Cognero Page 1
2 5. MEMORIES Zola has a picture frame that holds all of her school pictures. Each small opening is similar to the large opening in the center. If the center opening has an area of 33 square inches, what is the area of each small opening? The scale factor between the center opening and the small opening is, so the ratio of their areas is. For each pair of similar figures, find the area of the green figure. 7. The scale factor between the blue parallelogram and the green parallelogram is or, so the ratio of their areas is. The area of the green parallelogram is 240 ft 2. esolutions Manual - Powered by Cognero Page 2
3 9. The scale factor between the blue pentagon and the green pentagon is or, so the ratio of their areas is. The area of green pentagon is 672 cm 2. CCSS STRUCTURE For each pair of similar figures, use the given areas to find the scale factor of the blue to the green figure. Then find x. 11. The scale factor between the blue triangle and the green triangle is, so the ratio of their areas is. The scale factor is or. esolutions Manual - Powered by Cognero Page 3
4 13. The scale factor between the blue figure and the green figure is, so the ratio of their areas is. The scale factor is or. esolutions Manual - Powered by Cognero Page 4
5 15. BAKING Kaitlyn wants to use one of two regular hexagonal cake pans for a recipe she is making. The side length of the larger pan is 4.5 inches, and the area of the base of the smaller pan is 41.6 square inches. a. What is the side length of the smaller pan? b. The recipe that Kaitlyn is using calls for a circular cake pan with an 8-inch diameter. Which pan should she choose? Explain your reasoning. a. Area of the smaller pan = 41.6 We know that the area of the regular hexagon is is the number of sides. where s is the length of the side, A is the area, and n The interior angle of a hexagon is 60 and the apothem bisects the side length creating a triangle with a shorter leg that measures. The length of the apothem is therefore. b. The area of the circular pie pan is The area of the smaller pan is given 41.6 in 2, and the area of the larger pan can be determined using ratios. The larger pan should be used since the larger pan is closer to the area of the circular pan. esolutions Manual - Powered by Cognero Page 5
6 17. CHANGING DIMENSIONS A circle has a radius of 24 inches. a. If the area is doubled, how does the radius change? b. How does the radius change if the area is tripled? c. What is the change in the radius if the area is increased by a factor of x? a. Let A be the area. The new area is 2A. From theorem 11.1: If the area is doubled, the radius changes from 24 in. to 33.9 in. b. Let A be the area. The new area is 3A. From theorem 11.1: If the area is tripled, the radius changes from 24 in. to 41.6 in. c. Let A be the area. The new area is xa. From theorem 11.1: If the area changes by a factor of x, then the radius changes from 24 in. to 24 in. esolutions Manual - Powered by Cognero Page 6
7 COORDINATE GEOMETRY Find the area of each figure. Use the segment length given to find the area of a similar polygon. 19. Area of triangle JKL = 0.5(5)(6) or 15 The scale factor between and is or, so the ratio of their areas is. 21. esolutions Manual - Powered by Cognero Page 7
8 Area of a triangle = ½ bh Substitute. Area of triangle or 6. Area of triangle or 12. Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ADC = = 18 For the length of BC use the distance formula. B = ( 3, 7) and C = ( 1, 5). Here, BC is. The scale factor between quadrilateral and quadrilateral is or, so the ratio of their areas is. esolutions Manual - Powered by Cognero Page 8
9 34. In the figure, square WXYZ is inscribed in R. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point R, radius:, apothem:, central angle:, A square is a regular polygon with 4 sides. Thus, the measure of each central angle of square WXYZ is or 90. Find the area of the shaded region. Round to the nearest tenth. 35. The inscribed equilateral triangle can be divided into three congruent isosceles triangles with each central angle having a measure of or 120. Apothem is the height of isosceles triangle AOB, so it bisects AOB and. Thus, and AT = BT. Use trigonometric ratios to find the side length and apothem of the regular polygon. esolutions Manual - Powered by Cognero Page 9
10 Use the formula for finding the area of a regular polygon replacing a with OT and P with 3 AB to find the area of the triangle. Use the formula for the area of a circle replacing r with OB. Area of the shaded region = Area of the circle Area of the given triangle Substitute. esolutions Manual - Powered by Cognero Page 10
11 36. The radius of the circle is half the diameter or 3 feet. The area of the shaded region is the difference of the area of the rectangle and the area of the circle. Area of the shaded region = Area of the rectangle Area of the circle Substitute. 37. The regular pentagon can be divided into 5 congruent isosceles triangles with each central angle having a measure of or 72. esolutions Manual - Powered by Cognero Page 11
12 Apothem is the height of the isosceles triangle AOB, so it bisects AOB and. Thus, and AT = BT. Use trigonometric ratios to find the side length and apothem of the polygon. Use the formula for finding the area of a regular polygon replacing a with OT and P with 5 AB. Use the formula for the area of a circle replacing r with OB. Area of the shaded region = Area of the circle Area of the pentagon Substitute. esolutions Manual - Powered by Cognero Page 12
13 Find each measure. 38. m 5 By theorem 10.12, if two secants or chords intersect in the interior of a circle, then the measure of an angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle. This means that 39. m 6 By theorem 10.12, if two secants or chords intersect in the interior of a circle, then the measure of an angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle. We know that the sum of the arcs intercepted by 6 and its vertical angle will have a measure of 360 ( ) = 285. Theorem tells us that m 6 is half this: esolutions Manual - Powered by Cognero Page 13
14 40. m 7 Label the angle that forms a linear pair with as. By theorem 10.13, if a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one half the measure of its intercepted arc.. Since and are supplementary angles esolutions Manual - Powered by Cognero Page 14
15 41. State whether the figure has plane symmetry, axis symmetry, both, or neither. Consider rotating the cylinder about its vertical axis. Any degree of rotation will result in a symmetrical shape. Rotating the cylinder about any other axis will not provide symmetry. The cylinder has axis symmetric about its vertical axis. Now consider a vertical plane that cuts through the cylinder along its axis. If the cylinder is reflected through this plane, it will be symmetrical. If a horizontal plane cuts through the center of the cylinder, then the reflection will again produce symmetry. Any other plane, such as a vertical plane cutting through the right side, will not produce symmetry. The cylinder has plane symmetry about any vertical plane that cuts in line with the central axis, and also through the horizontal plane that cuts the cylinder in half. The cylinder has both axis and plane symmetry. esolutions Manual - Powered by Cognero Page 15
16 42. YEARBOOKS Tai resized a photograph that was 8 inches by 10 inches so that it would fit in a 4-inch by 4-inch area on a yearbook page. a. Find the maximum dimensions of the reduced photograph. b. What is the percent of reduction of the length? YEARBOOKS Tai resized a photograph that was 8 inches by 10 inches so that it would fit in a 4-inch by 4-inch area on a yearbook page. a. Find the maximum dimensions of the reduced photograph. b. What is the percent of reduction of the length? a. The photograph should be fit in to a 4-inch by 4-inch area so no side should exceed 4 in. The longest side, 10 in, should be reduced to 4 in. Let x be the other dimension of the reduced photograph. The scale ratio of corresponding side lengths will be equal: The maximum dimensions of the reduced photograph is 3.2 in. by 4 in. b. The side whose length is 10 inches should be reduced to 4, which is 40% of the original length. The original photo was reduced by 100% 40% = 60%. esolutions Manual - Powered by Cognero Page 16
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