Question

1. x + 5, 15, 5x + 1, 35

Explanation:

Answer:

To solve for \( x \), we use the Basic Proportionality Theorem (Thales' theorem) which states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally.

So, we set up the proportion:
\[
\frac{x + 5}{15} = \frac{5x + 1}{35}
\]

Cross - multiply:
\[
35(x + 5)=15(5x + 1)
\]

Expand both sides:
\[
35x+175 = 75x + 15
\]

Subtract \( 35x \) from both sides:
\[
175=40x + 15
\]

Subtract 15 from both sides:
\[
160 = 40x
\]

Divide both sides by 40:
\[
x = 4
\]