Chapter 6.7, Problem 15PPS

a.

To determine

To find: expression for the perimeter of ∆ABC

Answer to Problem 15PPS

Perimeter of $\Delta ABC=a+b+c$

Explanation of Solution

Given information: $\Delta ABC\sim \Delta XYZ,AB=c,BC=a,AC=b$

Calculation :

Perimeter refers to sum of all sides.

Perimeter of $\Delta ABC=a+b+c$

b.

To determine

To find : algebraic expressions for the measures of the sides of $\Delta XYZ$

Answer to Problem 15PPS

  $XY=dc,YZ=da,XZ=db$

Explanation of Solution

Given information: $\Delta ABC\sim \Delta XYZ$ , scale factor is represented by d.

Calculation :

As $\Delta ABC\text{and}\Delta XYZ$ are similar, so their sides are proportional.

  $\frac{XY}{AB}=\frac{YZ}{BC}=\frac{XZ}{AC}=d$

So,

  $XY=d(AB)=dc,YZ=d(BC)=da,XZ=d(AC)=db$

c.

To determine

To find : expression for the perimeter of ∆XYZ

Answer to Problem 15PPS

Perimeter of $\Delta XYZ=dc+da+db$

Explanation of Solution

Given information: $\Delta ABC\sim \Delta XYZ$ , scale factor is represented by d.

Calculation :

Perimeter is the sum of all sides.

Perimeter of $\Delta XYZ=dc+da+db$

d.

To determine

To factorize : expression $dc+da+db$

Answer to Problem 15PPS

  $dc+da+db=d(a+b+c)$

Explanation of Solution

Given information : expression is $dc+da+db$

Calculation :

  $dc+da+db=d(a+b+c)$

(according to distributive property $u(v+w)=uv+uw$ )

e.

To determine

To find : perimeter of $\Delta XYZ$

Answer to Problem 15PPS

Perimeter of $\Delta XYZ=24$ inches

Explanation of Solution

Given information : AB = 3 inches, BC = 4 inches, AC = 5 inches

Scale factor from $\Delta ABC$ to $\Delta XYZ$ is 2.

Calculation :

Perimeter of $\Delta XYZ=d(a+b+c)$

Put $d=2,c=3,a=4,b=5$

So,

Perimeter of $\Delta XYZ=2(3+4+5)=2(12)=24$ inches

f.

To determine

To find : Relation between the perimeters of similar figures and the scale factor

Answer to Problem 15PPS

Perimeter of $\Delta ABC=6d$

Perimeter of $\Delta XYZ=12d$

Explanation of Solution

Given information : AB = 3 inches, BC = 4 inches, AC = 5 inches

Scale factor from $\Delta ABC$ to $\Delta XYZ$ is 2.

Calculation :

Perimeter of $\Delta ABC=a+b+c=3+4+5=12$ inches

So,

Perimeter of $\Delta ABC=2(6)=6d$

Perimeter of $\Delta XYZ=2(12)=12d$