Question

1. the two figures are similar. solve for the variables (x, y, and z) and give the scale factor of the small figure to the big figure.
2. solve for x. *

*the two △s are similar.

Explanation:

Step1: Set up proportion for similar figures

For similar polygons, the ratios of corresponding sides are equal.
For problem 3:
The scale factor \(k\) (small to big) is \(\frac{9}{14}\) (since \( \frac{MK}{AD}=\frac{9}{14}\)).
For side \(x\): \(\frac{12}{28}=\frac{9}{14}=\frac{y}{42}=\frac{z}{14}\)
For side \(y\): \(\frac{y}{42}=\frac{9}{14}\), cross - multiply gives \(14y = 9\times42\), then \(y=\frac{9\times42}{14}=27\)
For side \(z\): \(\frac{z}{14}=\frac{9}{14}\), so \(z = 9\)
For problem 4:
Since the two triangles are similar, \(\frac{x}{x - 8}=\frac{21+14}{21}\) (using the ratio of corresponding sides). Cross - multiply: \(21x=(x - 8)\times35\)

Step2: Solve the equation

Expand \(21x=(x - 8)\times35\) to \(21x = 35x-280\)
Subtract \(21x\) from both sides: \(0=35x - 21x-280\), so \(14x=280\), then \(x = 20\)

Answer:

For problem 3: \(x = 18\), \(y = 27\), \(z = 9\), scale factor \(\frac{9}{14}\)
For problem 4: \(x = 20\)