Question

example 3: each pair of polygons is similar. find the value of x.
6.
7.
hint: \\(\frac{rp}{pm} = \frac{vz}{zw}\\)

Explanation:

Problem 6:

Step1: Identify corresponding sides

Since the polygons are similar, the ratios of corresponding sides are equal. From the diagram, the corresponding sides are \(x + 1\) and \(8\), \(3x + 1\) and \(20\). So we set up the proportion \(\frac{x + 1}{8}=\frac{3x + 1}{20}\).

Step2: Cross - multiply

Cross - multiplying gives us \(20(x + 1)=8(3x + 1)\).

Step3: Expand both sides

Expanding, we have \(20x+20 = 24x + 8\).

Step4: Solve for x

Subtract \(20x\) from both sides: \(20=4x + 8\). Then subtract 8 from both sides: \(12 = 4x\). Divide both sides by 4: \(x = 3\).

Problem 7:

Step1: Identify corresponding sides

Given the hint \(\frac{RP}{PM}=\frac{VZ}{ZW}\), from the diagram, \(RP = 3x-1\), \(PM=8x - 1\), \(VZ = 4\), \(ZW = 12\). So the proportion is \(\frac{3x-1}{8x - 1}=\frac{4}{12}\). Simplify \(\frac{4}{12}=\frac{1}{3}\), so \(\frac{3x-1}{8x - 1}=\frac{1}{3}\).

Step2: Cross - multiply

Cross - multiplying gives \(3(3x - 1)=8x - 1\).

Step3: Expand the left - hand side

Expanding, we get \(9x-3=8x - 1\).

Step4: Solve for x

Subtract \(8x\) from both sides: \(x-3=-1\). Then add 3 to both sides: \(x = 2\).

Answer:

s:
Problem 6: \(x = 3\)
Problem 7: \(x = 2\)