Question

solve for the variable or segment.
23.

1. find gh

Explanation:

Problem 23:

Step1: Identify the theorem (Angle Bisector Theorem)

The Angle Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. So, $\frac{18}{24}=\frac{x}{9}$.

Step2: Cross - multiply to solve for $x$

Cross - multiplying gives $24x = 18\times9$.

Step3: Calculate the right - hand side

$18\times9 = 162$, so $24x=162$.

Step4: Solve for $x$

Divide both sides by 24: $x=\frac{162}{24}=\frac{27}{4} = 6.75$.

Step1: Identify the theorem (Basic Proportionality Theorem - Thales' theorem for parallel lines)

When three or more parallel lines are cut by two transversals, the segments are proportional. So, $\frac{FG}{GH}=\frac{JK}{KL}$. Let $GH = x$. Then $\frac{28}{x}=\frac{30}{25}$.

Step2: Cross - multiply to solve for $x$

Cross - multiplying gives $30x=28\times25$.

Step3: Calculate the right - hand side

$28\times25 = 700$, so $30x = 700$.

Step4: Solve for $x$

Divide both sides by 30: $x=\frac{700}{30}=\frac{70}{3}\approx23.33$.

Answer:

$x = \frac{27}{4}$ (or $6.75$)

Problem 24: