Perimeters and Areas of Similar Figures Geometry Perimeters and Areas of Similar Figures

Is this statement True or False? The picture is twice as big as our smaller TV. Screen Size: 6  9 Screen Size: 12  18 Area: 54 Area: 216 April 27, 2017

Goals Compare perimeters and areas of similar figures. Use perimeters and areas of similar figures to solve problems. April 27, 2017

Problem 6 24 Draw a rectangle that measures 3  2. The area is ? Double the lengths of the sides and draw a rectangle that measures 6  4. The area is ? 24 April 27, 2017

Example continued When the lengths of the sides were doubled, did the area double? NO! 6 24 April 27, 2017

? ? So what happened? 6 24 3 The ratio of the sides: 2 The ratio of the areas: 6 ? 24 4 April 27, 2017

Notice: 6 24 3 The ratio of the sides: 2 The ratio of the areas: 6 4 April 27, 2017

Theorem 10.5 The Area of Similar Polygons If two polygons are similar with lengths of corresponding sides in the ratio a:b, then the ratio of their areas is a2:b2. Another way of saying it: the ratio of the areas is the square of the ratio of the sides. April 27, 2017

Example 1 ABCDE ~ MNOPQ 4 6 O P D C B N E Q A M Find the ratio of the areas of the figures. April 27, 2017

Example 2 These figures are similar. The area of the smaller one is 20. Find the area of the larger figure. 4 Ratio of the sides: Ratio of their areas: 10 April 27, 2017

Example 2 continued 20 125 x Write the proportion and solve: 4 10 Ratio of sides 2:5 Ratio of areas 4:25 April 27, 2017

Your Turn 5 These figures are similar. The area of the larger one is 320. Find the area of the smaller figure. ? 8 320 April 27, 2017

Solution 5 125 ? 8 320 April 27, 2017

Now do you see how it works? Do NOT write Ratio of sides. 5(320) = 1600 and 8(125)=1000 Instead write Ratio of SQUARE of sides. 25(320) = 8000 and 64(125) = 8000 April 27, 2017

Try it again… These figures are similar. Find the area of the smaller figure if the area of the larger one is 720. 5 12 A = 720 April 27, 2017

Solution 5 12 A = 720 April 27, 2017

Problem If the figures are similar, what is the length of the side marked x? 4 A = 36 x A = 100 Remember: the ratio of the areas is the square of the ratios of the sides. Or, the ratio of the sides is the square root of the ratios of the areas. April 27, 2017

Problem Solution 4 A = 36 x A = 100 The ratio of the sides is the square root of the ratios of the areas. April 27, 2017

Here’s nothing surprising: If two polygons are similar, the ratio of the perimeters is equal to the ratio of the sides. 4 6 2 3 Perimeter = 10 Perimeter = 20 April 27, 2017

Example ABCD ~ DEFG. AB = 8 and DE = 12. The perimeter of ABCD is 20. What is the perimeter of DEFG? Solution: April 27, 2017

Summary The ratio of the areas of similar figures is the square of the ratio of the sides. The ratio of the perimeters of similar figures is the same as the ratio of the sides. The ratio of the sides in two similar figures is the square root of the ratio of the areas. April 27, 2017

Do you get it? Do these problems. 1. Figure CAT is similar to BAT. CA = 10 and BA = 15. What is the ratio of the sides? What is the ratio of the perimeters of CAT to BAT? What is the ratio of the areas of CAT to BAT? 2/3 2/3 4/9 April 27, 2017

2. Figure RATS is similar to DOGS. RA = 12 and DO = 30. What is the ratio of the perimeters of RATS to DOGS? What is the ratio of the areas of RATS to DOGS? April 27, 2017

3. Two figures, ABCDE and XYZWS are similar 3. Two figures, ABCDE and XYZWS are similar. The perimeter of ABCDE is and the perimeter of XYZWS is 18. What is the ratio of corresponding sides? (What is the scale factor?) April 27, 2017

4. JKL ~ RST. The area of JKL is 25 and the area of RST is 64 4. JKL ~ RST. The area of JKL is 25 and the area of RST is 64. What is the ratio of their perimeters? April 27, 2017

Ratio of Areas: Ratio of Perimeters: 5. HOT ~ DOG. The area of HOT is 100 and the area of DOG is 9. The perimeter of HOT is 40. What is the perimeter of DOG? Ratio of Areas: Ratio of Perimeters: April 27, 2017

4. HOT ~ DOG. The area of HOT is 100 and the area of DOG is 9 4. HOT ~ DOG. The area of HOT is 100 and the area of DOG is 9. The perimeter of HOT is 40. What is the perimeter of DOG? Ratio of Perimeters: April 27, 2017