Question

solve for x. the polygons in each pair are similar
7)
find the missing length indicated.
9)
a) 15
b) 25
c) 35
d) 10

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides proportionally. Let the missing length be \(x\). The ratio of the segments of one side is equal to the ratio of the segments of the other side.
$$\frac{9}{9 + 15}=\frac{x}{x + 15}$$
But a more straightforward way is using the similarity - related proportion for the two - part side. Since the line divides the sides proportionally, we can set up the proportion based on the side lengths.
We know that \(\frac{9}{9 + 15}=\frac{\text{segment of the divided side}}{\text{whole side}}\). Another approach is using the ratio of the non - divided side and the divided side.
Let's use the ratio of the side with length \(9\) and \(9+15 = 24\) and the side with the unknown \(x\) and \(x + 15\). But an easier proportion is based on the fact that \(\frac{9}{24}=\frac{\text{segment}}{\text{whole}}\). However, if we consider the ratio of the two parts of the non - parallel sides.
We have \(\frac{9}{15}=\frac{15}{x}\) (by the property of similar triangles formed by the parallel line, the ratio of corresponding sides is equal). Cross - multiply:

Step2: Cross - multiply the proportion

Cross - multiplying the proportion \(\frac{9}{15}=\frac{15}{x}\) gives \(9x=15\times15\). Then \(9x = 225\).

Step3: Solve for \(x\)

Divide both sides of the equation \(9x = 225\) by \(9\). So \(x=\frac{225}{9}=25\).

Answer:

B. 25