Geometry unit 10.4
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This document discusses the effects of changing dimensions on the perimeter and area of geometric figures. It provides examples of how multiplying or doubling individual dimensions, such as the height of a rectangle or radius of a circle, affects the perimeter and area. It also examines how proportionally changing multiple dimensions, such as doubling both the base and height of a rectangle, impacts perimeter and area. The key points are that individually changing a dimension proportionally changes the perimeter or area, while proportionally changing multiple dimensions results in a similar figure. The document aims to build understanding of these relationships through worked examples and practice problems.
Geometry unit 10.4
- 1. UNIT 10.4 PERIMETER AND AREAS OFUNIT 10.4 PERIMETER AND AREAS OF SIMILAR FIGURESSIMILAR FIGURES
- 2. Warm Up Find the area of each figure. Give exact answers, using π if necessary. 1. a square in which s = 4 m 2. a circle in which r = 2 ft 3. ABC with vertices A(–3, 1), B(2, 4), and C(5, 1) 16 m2 4π ft2 12 units2
- 3. Describe the effect on perimeter and area when one or more dimensions of a figure are changed. Apply the relationship between perimeter and area in problem solving. Objectives
- 4. In the graph, the height of each DVD is used to represent the number of DVDs shipped per year. However as the height of each DVD increases, the width also increases, which can create a misleading effect.
- 5. Describe the effect of each change on the area of the given figure. Example 1: Effects of Changing One Dimension The height of the triangle is multiplied by 6. original dimensions: multiply the height by 6: Notice that 180 = 6(30). If the height is multiplied by 6, the area is also multiplied by 6. = 30 in2 = 180 in2
- 6. Example 1B: Effects of Changing One Dimension original dimensions: The diagonal SU of the kite with vertices R(2, 2), S(4, 0), T(2, –2), and U(–5,0) is multiplied by .
- 7. Check It Out! Example 1 The height of the rectangle is tripled. Describe the effect on the area. A = bh = (7)(4) A = bh = (7)(12) = 28 ft2 = 84 ft2 Notice that 84 = 3(28). If the height is multiplied by 3, the area is tripled. original dimensions: triple the height:
- 8. If the radius of a circle or the side length of a square is changed, the size of the entire figure changes proportionally. Helpful Hint
- 9. Describe the effect of each change on the perimeter or circumference and the area of the given figures. Example 2A: Effects of Changing Dimensions Proportionally The base and height of a rectangle with base 4 ft and height 5 ft are both doubled.
- 10. Example 2A Continued P = 2(8) + 2(10) = 36 ft 2(4) = 8; 2(5) = 10 A = (8)(10) = 80 ft2 2(18) = 38 dimensions doubled: The perimeter is multiplied by 2. The area is multiplied by 22 , or 4. 4(20) = 80 P = 2(4) + 2(5) = 18 ft P = 2b + 2h A = (4)(5) = 20 ft2 A = bh original dimensions:
- 11. Example 2B: Effects of Changing Dimensions Proportionally C = 2π(10) = 20π cm C = 2πr A = π(10)2 = 100π cm2 original dimensions: C = 2π(2) = 4π cm A = π(2)2 = 4π cm2 A = πr2 The radius of J is multiplied by . dimensions multiplied by .
- 12. Example 2B Continued The area is multiplied by The circumference is multiplied by .
- 13. Check It Out! Example 2 The base and height of the triangle with vertices P(2, 5), Q(2, 1), and R(7, 1) are tripled. Describe the effect on its area and perimeter. The perimeter is tripled, and the area is multiplied by 9. original dimensions: dimensions tripled:
- 14. When the dimensions of a figure are changed proportionally, the figure will be similar to the original figure.
- 15. Example 3A: Effects of Changing Area A circle has a circumference of 32π in. If the area is multiplied by 4, what happens to the radius? πr2 = 1024π r2 = 1024 r = √1024 = 32 Set the new area equal to πr2 . Divide both sides by π. Take the square root of both sides and simplify. and the area is A = πr2 = 256π in2 . If the area is multiplied by 4, the new area is 1024π in2 . The original radius is Notice that 32π = 2(16π). The radius is multiplied by 2.
- 16. Example 3B: Effects of Changing Area Let s be a side length of an equilateral triangle. Draw a segment that bisects the top angle and the base to form a 30-60-90 triangle. An equilateral triangle has a perimeter of 21m. If the area is multiplied by , what happens to the side length? .
- 17. Example 3B Continued The length of each side is , and the area of the equilateral triangle is If the area is multiplied by , the new area is
- 19. Check It Out! Example 3 A square has a perimeter of 36 mm. If the area is multiplied by , what happens to the side length?
- 20. Explain why the graph is misleading. Example 4: Entertainment Application The height of the bar representing sales in 2000 is about 2.5 times the height of the bar representing sales in 2003.
- 21. Example 4 Continued This means that the area of the bar multiplied by about 2.52 , or 6.25, so the area of the larger bar is about 6.25 times the area of the smaller bar. The graph gives the misleading impression that the number of sales in 2003 decreased by 6 times the sales in 2000, but the decrease was actually closer to 2.5 times.
- 22. Check It Out! Example 4 Use the information in example 4 to create a version of the graph that is not misleading.
- 23. Lesson Quiz: Part I Describe the effect of each change on the area of the given figure. The area is multiplied by 8. 1. The base length of the rectangle is multiplied by 8. The area is multiplied by 9. 2. The radius of the circle is tripled.
- 24. Lesson Quiz: Part II The side length is doubled. 3. A square has an area of 49 cm2 . If the area is quadrupled, what happens to the side length? 4. Rob had a 10 ft by 12 ft wall painted. For a wall twice as wide, the painter charged him twice as much. Is this reasonable? Explain. Yes; the second wall has twice the area of the first wall.