Exercise 21 Page 642

If the scale factor of two similar figures is ab, then the scale factor of the perimeters is also ab, and the scale factor of their areas is a^2b^2.

Scale Factor	Ratio of the Perimeters	Ratio of the Areas
a:b or a/b	a:b or a/b	a^2:b^2 or a^2/b^2

From the exercise, we know that the ratio between the perimeters of two similar triangles is 1:3= 13. Therefore, the scale factor of the two triangles is also 13. To find the ratio of their areas, we will raise 13 to the second power.

a/b=1/3

RaiseEqn

LHS^2=RHS^2

(a/b)^2=(1/3)^2

PowQuot

(a/b)^m=a^m/b^m

a^2/b^2=1^2/3^2

CalcPow

Calculate power

a^2/b^2=1/9

The ratio of the areas is 19. Finally, let x be the area of the smaller triangle. We will write and solve a proportion using the ratio of the areas and the area of the larger triangle, which is 27ft^2.

1/9=x/27

▼

Solve for x

CrossMult

Cross multiply

1(27)=9x

IdPropMult

Identity Property of Multiplication

27=9x

DivEqn

.LHS /9.=.RHS /9.

3=x

RearrangeEqn

Rearrange equation

x=3

The area of the smaller figure is 3ft^2.