Question

quadrilateral mnop is dilated by a scale factor of \frac{3}{4} to form quadrilateral mnop. what is the measure of side op?

Explanation:

Step1: Recall the dilation formula

If a figure is dilated by a scale factor \(k\), the length of a side of the original figure \(x\) and the length of the corresponding side of the dilated figure \(x'\) are related by the formula \(x' = kx\). Here, \(k=\frac{3}{4}\), and we want to find \(x\) (length of \(OP\)) given \(x'\) (length of \(O'P'\)). Rearranging the formula gives \(x=\frac{x'}{k}\).

Step2: Substitute the values

We know that \(x' = 15\) (length of \(O'P'\)) and \(k=\frac{3}{4}\). Then \(x=\frac{15}{\frac{3}{4}}\). Using the rule for dividing by a fraction \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}\), we have \(x = 15\times\frac{4}{3}\).

Step3: Calculate the result

\(15\times\frac{4}{3}=\frac{15\times4}{3}\). Since \(15\div3 = 5\), then \(5\times4=20\).

Answer:

\(20\)