Question

first determine if the triangles are similar. then find a and b.

Explanation:

Step1: Determine similarity

Since two angles of one triangle are equal to two angles of the other triangle (AA - Angle - Angle similarity criterion), the triangles are similar.

Step2: Set up proportion for \(a\)

The ratio of corresponding sides is equal. For the sides \(6\) and \(6.4\) (corresponding sides), and \(7\) and \(a\) (corresponding sides).
The proportion is \(\frac{6}{6.4}=\frac{7}{a}\).
Cross - multiply: \(6a = 6.4\times7\).
\(6a=44.8\).
\(a=\frac{44.8}{6}=\frac{448}{60}=\frac{112}{15}\approx7.47\).

Step3: Set up proportion for \(b\)

Using the ratio \(\frac{6}{6.4}=\frac{8}{b}\).
Cross - multiply: \(6b = 6.4\times8\).
\(6b = 51.2\).
\(b=\frac{51.2}{6}=\frac{512}{60}=\frac{128}{15}\approx8.53\).

Answer:

The triangles are similar. \(a = \frac{112}{15}\approx7.47\) and \(b=\frac{128}{15}\approx8.53\)