Question

the diagram shows a dilation with center ( o ) and scale factor ( \frac{3}{2} ).

what is the distance between ( o ) and ( q )?

( square mathrm{cm} )

Explanation:

Step1: Recall the dilation formula

If a dilation has center \(O\) and scale factor \(k\), and a point \(Q\) is dilated to \(Q'\), then \(OQ'=k\times OQ\). Here \(k = \frac{3}{2}\) and \(OQ'=6\) cm.

Step2: Solve for \(OQ\)

From \(OQ'=k\times OQ\), we can rewrite it as \(OQ=\frac{OQ'}{k}\). Substitute \(OQ' = 6\) and \(k=\frac{3}{2}\) into the formula: \(OQ=\frac{6}{\frac{3}{2}}\).
Using the rule \(\frac{a}{\frac{b}{c}}=a\times\frac{c}{b}\), we have \(OQ = 6\times\frac{2}{3}\).
Calculate \(6\times\frac{2}{3}\): \(6\div3\times2 = 2\times2=4\).

Answer:

\(4\)