Question

triangle abc is similar to xyz. if a = 24, what are x and y?

Explanation:

Step1: Set up proportion for similar - triangles

Since \(\triangle ABC\sim\triangle XYZ\), the ratios of corresponding sides are equal. We have \(\frac{AB}{XY}=\frac{AC}{XZ}=\frac{BC}{YZ}\). Let's use the ratio \(\frac{AB}{XY}=\frac{AC}{XZ}\). We know \(AB = 15\), \(AC = 21\), \(XY=10\), and \(XZ = x\). So, \(\frac{15}{10}=\frac{21}{x}\).

Step2: Cross - multiply to solve for \(x\)

Cross - multiplying the proportion \(\frac{15}{10}=\frac{21}{x}\) gives us \(15x=21\times10\). Then \(15x = 210\), and \(x=\frac{210}{15}=14\).

Step3: Set up another proportion to solve for \(y\)

Use the ratio \(\frac{AB}{XY}=\frac{BC}{YZ}\). We know \(AB = 15\), \(BC = 24\), \(XY = 10\), and \(YZ=y\). So, \(\frac{15}{10}=\frac{24}{y}\).

Step4: Cross - multiply to solve for \(y\)

Cross - multiplying the proportion \(\frac{15}{10}=\frac{24}{y}\) gives us \(15y=24\times10\). Then \(15y = 240\), and \(y=\frac{240}{15}=16\).

Answer:

\(x = 14\), \(y = 16\)