Question

given \\( \triangle lmk \sim \triangle ghj \\) and the scale factor is \\( \frac { 2 } { 3 } \\), find the length of each side of \\( \triangle ghj \\)

hj=

gh=

jg=

Explanation:

Step1: Determine the correspondence of sides

Since \(\triangle LMK\sim\triangle GHJ\) and the scale factor is \(\frac{2}{3}\), we have the following correspondences: \(HJ\) corresponds to \(LK\), \(GH\) corresponds to \(ML\), and \(JG\) corresponds to \(KM\).

Step2: Calculate the length of \(HJ\)

Given \(LK = 24\). Using the formula for similar - triangles (if the scale factor of \(\triangle LMK\) to \(\triangle GHJ\) is \(\frac{2}{3}\), then \(\frac{LK}{HJ}=\frac{2}{3}\)). Let \(x = HJ\), then \(\frac{24}{x}=\frac{2}{3}\). Cross - multiply: \(2x=24\times3\), \(2x = 72\), \(x=\frac{72}{2}=36\).

Step3: Calculate the length of \(GH\)

Given \(ML = 12\). Let \(y = GH\). Since \(\frac{ML}{GH}=\frac{2}{3}\), then \(\frac{12}{y}=\frac{2}{3}\). Cross - multiply: \(2y=12\times3\), \(2y = 36\), \(y = 18\).

Step4: Calculate the length of \(JG\)

Given \(KM = 30\). Let \(z = JG\). Since \(\frac{KM}{JG}=\frac{2}{3}\), then \(\frac{30}{z}=\frac{2}{3}\). Cross - multiply: \(2z=30\times3\), \(2z = 90\), \(z=\frac{90}{2}=45\).

Answer:

\(HJ = 36\)
\(GH = 18\)
\(JG = 45\)