Question

1. solve for the variables a = ___ b = _ c = ___

Explanation:

Step1: Use similar - triangle property

When we have parallel lines in a triangle, we can use the property of similar - triangles. The ratios of corresponding sides of similar triangles are equal.
For the left - hand side of the large triangle, we consider the ratios of the segments formed by the parallel lines. Let's consider the first set of similar triangles for finding \(a\).
We have \(\frac{a}{a + 3+6+7.5}=\frac{6}{6 + 8 + c}\). But we can also use the ratio of the first two small similar triangles on the left - hand side. The ratio of the sides of the first two small similar triangles gives \(\frac{a}{a + 3}=\frac{6}{6 + 8}\). Cross - multiply: \(8a=18\), so \(a = 2.25\).

Step2: Find \(b\) using similar - triangle ratio

Using the similar - triangle property for the right - hand side of the triangle with the first two small similar triangles. The ratio of the sides of the first two small similar triangles on the right - hand side is \(\frac{b}{b + 8}=\frac{3}{3+6}\). Cross - multiply: \(3(b + 8)=9b\), \(3b+24 = 9b\), \(6b=24\), so \(b = 4\).

Step3: Find \(c\) using similar - triangle ratio

We consider the large and medium - sized triangles formed by the parallel lines. The ratio of the left - hand side segments of the large and medium - sized triangles is the same as the ratio of the right - hand side segments.
The ratio of the left - hand side segments of the large and medium - sized triangles is \(\frac{3 + 6+7.5}{3 + 6}=\frac{c + 8}{8}\). First, \(\frac{16.5}{9}=\frac{c + 8}{8}\). Cross - multiply: \(16.5\times8=9(c + 8)\), \(132=9c+72\), \(9c = 60\), \(c = 10\).

Answer:

\(a = 2.25\), \(b = 4\), \(c = 10\)