Question

1. the polygons are similar, but not necessarily drawn to scale. find the values of x and y.
2. given that the pair of triangles are similar, solve for the missing variable. round to the nearest tenth. draw the triangles separately to start this problem.

Explanation:

Problem 11

Step1: Find the scale factor

Since the polygons are similar, the ratio of corresponding sides is equal. The ratio of the vertical sides is \(\frac{16}{12}=\frac{4}{3}\).

Step2: Solve for \(x\)

Set up the proportion for the top - side: \(\frac{x - 1}{6}=\frac{4}{3}\). Cross - multiply: \(3(x - 1)=4\times6\), \(3x-3 = 24\), \(3x=27\), \(x = 9\).

Step3: Solve for \(y\)

Set up the proportion for the non - parallel side: \(\frac{y + 1}{21}=\frac{4}{3}\). Cross - multiply: \(3(y + 1)=4\times21\), \(3y+3 = 84\), \(3y=81\), \(y = 27\).

Step1: Set up the proportion

Since the triangles are similar, \(\frac{28}{36}=\frac{2x + 4}{2x+4 + x}\). Simplify the right - hand side denominator: \(2x+4 + x=3x + 4\). So the proportion is \(\frac{28}{36}=\frac{2x + 4}{3x + 4}\).

Step2: Cross - multiply

\(28(3x + 4)=36(2x + 4)\). Expand: \(84x+112 = 72x + 144\).

Step3: Solve for \(x\)

Subtract \(72x\) from both sides: \(84x-72x+112=72x-72x + 144\), \(12x+112 = 144\). Subtract \(112\) from both sides: \(12x=144 - 112\), \(12x = 32\). Divide by \(12\): \(x=\frac{32}{12}\approx2.7\).

Answer:

\(x = 9\), \(y = 27\)

Problem 12