Question

triangle grs is similar to triangle tuy. find the measure of side tu. figures are not drawn to scale.

Explanation:

Step1: Set up proportion

Since the triangles are similar, the ratios of corresponding sides are equal. Let the sides of the first - triangle be \(a = 9\), \(b = 10\), \(c = 7\) and the sides of the second - triangle be \(A=x\), \(B = 6\), \(C = y\). We can assume that \(\frac{a}{A}=\frac{b}{B}=\frac{c}{C}\). Let's assume \(\frac{9}{x}=\frac{7}{6}\) (assuming the correct correspondence of sides).

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

Cross - multiplying the proportion \(\frac{9}{x}=\frac{7}{6}\) gives us \(7x=9\times6\). Then \(7x = 54\), and \(x=\frac{54}{7}\approx7.71\). If we assume \(\frac{9}{6}=\frac{10}{x}\), cross - multiplying gives \(9x = 6\times10\), so \(9x=60\) and \(x=\frac{60}{9}=\frac{20}{3}\approx6.67\). If we assume \(\frac{9}{6}=\frac{7}{x}\), cross - multiplying gives \(9x = 6\times7\), so \(9x = 42\) and \(x=\frac{42}{9}=\frac{14}{3}\approx4.67\). But if we assume the proportion \(\frac{10}{6}=\frac{9}{x}\), cross - multiplying gives \(10x=6\times9\), so \(10x = 54\) and \(x = 5.4\). If we assume \(\frac{10}{6}=\frac{7}{x}\), cross - multiplying gives \(10x=6\times7\), so \(10x = 42\) and \(x = 4.2\). Let's assume the correct proportion is \(\frac{9}{x}=\frac{10}{6}\).
Cross - multiply: \(10x=9\times6\).
\(10x = 54\), then \(x = 5.4\).

Answer:

5.4