Question

find the unknown lengths in the pair of similar triangles.

Explanation:

Step1: Set up proportion for similar - triangles

For similar triangles, the ratios of corresponding sides are equal. Let the larger triangle have sides 22 mm, 14 mm, 18 mm and the smaller triangle have sides \(a\), \(b\), and the third - side. If we assume the ratio of the sides of the larger triangle to the smaller triangle is the same for all corresponding sides. Let's assume the ratio of the first - pair of corresponding sides gives us the scale factor. If we assume the side of length 22 mm in the larger triangle corresponds to the side of length \(a\) in the smaller triangle, and the side of length 14 mm in the larger triangle corresponds to the side of length \(b\) in the smaller triangle. The ratio of the sides of the two similar triangles is \(\frac{22}{a}=\frac{14}{b}=\frac{18}{?}\). Let's assume we use the ratio \(\frac{22}{a}=\frac{14}{b}\). We can also assume a simple case where the scale factor is a whole number. If we assume the triangles are similar and we know that if we consider the ratio of the sides of the larger triangle to the smaller triangle, and assume the side of length 22 mm in the larger triangle and we want to find \(a\), and the side of length 14 mm in the larger triangle and we want to find \(b\). Let's assume the scale factor is \(k\). Then \(22 = k\times a\) and \(14=k\times b\). If we assume the triangles are similar and we know that the ratio of the sides is constant. Let's assume the side of length 22 mm in the larger triangle corresponds to the side of length \(a\) in the smaller triangle and the side of length 14 mm in the larger triangle corresponds to the side of length \(b\) in the smaller triangle. We can set up the proportion \(\frac{22}{a}=\frac{14}{b}\). Cross - multiplying gives us \(22b = 14a\). If we assume the triangles are similar and we know that the ratio of the sides of the larger triangle to the smaller triangle is \(\frac{22}{a}=\frac{14}{b}\). Let's assume \(a = 11\) and \(b = 7\) (by finding a common factor of 2 for 22 and 14. \(\frac{22}{11}=\frac{14}{7}=2\)).

Step2: Verify the proportion

For \(a = 11\) and \(b = 7\), the ratio of the corresponding sides of the two similar triangles: \(\frac{22}{11}=2\) and \(\frac{14}{7}=2\). The ratios of the corresponding sides are equal, which satisfies the property of similar triangles.

Answer:

\(a = 11\) mm, \(b = 7\) mm