Question

date:12-8-25
ty theorems worksheet

1. find x.

32 25
x
15

Explanation:

Step1: Identify the theorem

This is a triangle with a midline (or a line parallel to the base), so we use the Basic Proportionality Theorem (Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Let the length of the segment on the left side be \( x \), the total left side is 32, the upper segment on the right side is 25, and the lower segment on the right side is 15. So the ratio of the segments on the left side should be equal to the ratio of the segments on the right side.
The proportion is \( \frac{x}{32 - x}=\frac{25}{15} \) (Wait, actually, the correct proportion is based on the segments created by the parallel line. Let's re - establish: The two sides of the triangle are divided into lengths. The side with length 32 is divided into \( x \) and \( 32 - x \)? No, wait, looking at the diagram, the right side has lengths 25 (upper) and 15 (lower), and the left side has total length 32, with the upper part being \( x \). Since the line is parallel to the base, by Thales' theorem, \( \frac{x}{32}=\frac{25}{25 + 15} \)

Step2: Solve the proportion

First, simplify the right - hand side: \( \frac{25}{25+15}=\frac{25}{40}=\frac{5}{8} \)
So we have the equation \( \frac{x}{32}=\frac{5}{8} \)
Cross - multiply: \( 8x=32\times5 \)
\( 8x = 160 \)
Divide both sides by 8: \( x=\frac{160}{8}=20 \)

Answer:

\( x = 20 \)