Question

1. find the value of x. 10. solve for the variables

Explanation:

Step1: Use the property of similar - triangles

For the first triangle, if two triangles are similar, the ratios of their corresponding sides are equal. We have the proportion $\frac{6}{9}=\frac{4}{2x - 10}$.

Step2: Cross - multiply

Cross - multiplying the proportion $\frac{6}{9}=\frac{4}{2x - 10}$ gives us $6(2x - 10)=4\times9$.

Step3: Expand and solve for x

Expand the left - hand side: $12x-60 = 36$. Add 60 to both sides: $12x=36 + 60=96$. Divide both sides by 12: $x = 8$.

For the second set of similar triangles:

Step1: Use the property of similar - triangles for the first pair of parallel lines

For the lines with lengths $a$ and 6, and the corresponding sides of the large and small triangles, we have the proportion $\frac{a}{6}=\frac{3}{6}$. Cross - multiplying gives $6a=3\times6$, so $a = 3$.

Step2: Use the property of similar - triangles for the second pair of parallel lines

For the lines with lengths $b$ and 8, consider the similar triangles formed. The ratio of the corresponding sides of the similar triangles gives $\frac{b}{8}=\frac{3 + 6}{6+6}$. Since $\frac{3 + 6}{6+6}=\frac{9}{12}=\frac{3}{4}$, then $4b=8\times3$, so $b = 6$.

Step3: Use the property of similar - triangles for the third pair of parallel lines

For the lines with lengths $c$ and the whole side, consider the large similar triangles. The ratio of the corresponding sides of the similar triangles gives $\frac{c}{8}=\frac{3+6 + 7.5}{6+6+7.5}$. First, $3+6 + 7.5=16.5$ and $6+6+7.5 = 19.5$. The proportion is $\frac{c}{8}=\frac{16.5}{19.5}=\frac{11}{13}$. Cross - multiplying gives $13c=8\times11$, so $c=\frac{88}{13}\approx6.77$.

Answer:

$x = 8$
$a = 3$
$b = 6$
$c=\frac{88}{13}$