Question

sameer draws triangles abc and xyz. then he writes the quotients of the corresponding side lengths as fractions. sameer says that △xyz is a dilation of △abc. is he correct? explain your reasoning.
solution
\\( \frac { a b } { x y } = \frac { 5 } { 20 } \\)
\\( \frac { b c } { y z } = \frac { 7 } { 28 } \\)
\\( \frac { a c } { x z } = \frac { 8 } { 32 } \\)

Explanation:

Step1: Simplify the fractions

Simplify \(\frac{AB}{XY}=\frac{5}{20}\), \(\frac{BC}{YZ}=\frac{7}{28}\), \(\frac{AC}{XZ}=\frac{8}{32}\).
For \(\frac{5}{20}\), divide numerator and denominator by \(5\), we get \(\frac{5\div5}{20\div5}=\frac{1}{4}\).
For \(\frac{7}{28}\), divide numerator and denominator by \(7\), we get \(\frac{7\div7}{28\div7}=\frac{1}{4}\).
For \(\frac{8}{32}\), divide numerator and denominator by \(8\), we get \(\frac{8\div8}{32\div8}=\frac{1}{4}\).

Step2: Check the similarity condition

Since \(\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}=\frac{1}{4}\), by the Side - Side - Side (SSS) similarity criterion, \(\triangle ABC\sim\triangle XYZ\).
A dilation is a transformation that changes the size of a figure but not its shape. When two triangles are similar (in this case by SSS similarity as all corresponding side ratios are equal), one triangle is a dilation of the other.

Answer:

Sameer is correct. Because \(\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}=\frac{1}{4}\), by SSS similarity \(\triangle ABC\sim\triangle XYZ\), and since they are similar (same shape, different size), \(\triangle XYZ\) is a dilation of \(\triangle ABC\).