Question

7-19 the polygons below are similar. what is the scale factor? find the missing side lengths.
diagram: first polygon with sides 6 ft, 5.7 ft, 12 ft, ( x ); second polygon with sides 4 ft, ( z ), 6 ft, ( y )
7-27 solve the system of equations below.
( y = 3x + 1 )
( x + 2y = -5 )

Explanation:

7-19 Solution:

Step1: Find the scale factor

The corresponding sides are 6 ft (left side of first polygon) and 4 ft (left side of second polygon). Scale factor \( k=\frac{4}{6}=\frac{2}{3} \).

Step2: Find \( z \)

Corresponding side to 5.7 ft is \( z \). Using scale factor: \( z = 5.7\times\frac{2}{3}=3.8 \) ft.

Step3: Find \( x \)

Corresponding side to \( x \) is 6 ft. Let's find \( x \) from the other side. Wait, first polygon's bottom side is 12 ft, second's? Wait, no, the second polygon's slant side is 6 ft? Wait, no, first polygon: left side 6, top 5.7, bottom 12, slant \( x \). Second: left 4, top \( z \), slant 6, bottom \( y \). Wait, maybe I mixed. Wait, scale factor from first to second is \( \frac{4}{6}=\frac{2}{3} \). So for \( x \): second's slant is 6, so \( 6 = x\times\frac{2}{3}\Rightarrow x = 6\times\frac{3}{2}=9 \) ft.

Step4: Find \( y \)

Corresponding to 12 ft: \( y = 12\times\frac{2}{3}=8 \) ft.

Step1: Substitute \( y = 3x + 1 \) into \( x + 2y=-5 \)

Substitute \( y \) in the second equation: \( x + 2(3x + 1)=-5 \).

Step2: Simplify and solve for \( x \)

Expand: \( x + 6x + 2=-5 \Rightarrow 7x + 2=-5 \). Subtract 2: \( 7x=-7 \Rightarrow x=-1 \).

Step3: Find \( y \) using \( x=-1 \)

Substitute \( x=-1 \) into \( y = 3x + 1 \): \( y = 3(-1)+1=-3 + 1=-2 \).

Answer:

Scale factor: \( \frac{2}{3} \); \( z = 3.8 \) ft, \( x = 9 \) ft, \( y = 8 \) ft.

7-27 Solution: