Question

solve for x. figures are not necessarily drawn to scale.

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle ONP \) and \( \triangle LNM \) are similar by the Vertical Angles Theorem and the AA (Angle - Angle) Similarity Criterion (since the vertical angles at \( N \) are equal and the other angles are corresponding angles formed by the transversal intersecting the two lines, so they are equal). For similar triangles, the ratios of corresponding sides are equal.
The corresponding sides are \( ON = 20 \), \( LN = 32 \), \( OP=x \), and \( LM = 24 \). So we can set up the proportion:
\( \frac{ON}{LN}=\frac{OP}{LM} \)

Step2: Substitute the Known Values

Substitute \( ON = 20 \), \( LN = 32 \), \( OP=x \), and \( LM = 24 \) into the proportion:
\( \frac{20}{32}=\frac{x}{24} \)

Step3: Solve for \( x \)

Cross - multiply to solve for \( x \). Cross - multiplying gives us \( 32x=20\times24 \).
First, calculate \( 20\times24 = 480 \). Then we have the equation \( 32x = 480 \).
To solve for \( x \), divide both sides of the equation by 32:
\( x=\frac{480}{32} \)
Simplify \( \frac{480}{32}\): divide numerator and denominator by 16, we get \( \frac{30}{2}=15 \).

Answer:

\( x = 15 \)