Question

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Explanation:

To solve the problem of finding the value of \( x \) in the first example (finding \( x \) where the angle bisector creates two right triangles with equal segments), we use the Angle Bisector Theorem (or the property that the distances from a point on the angle bisector to the sides of the angle are equal).

Step 1: Set up the equation

Since the segments are equal (from the angle bisector theorem for right triangles), we set the two expressions equal:

$$ 4x + 30 = 9x - 5 $$

Step 2: Solve for \( x \)

Subtract \( 4x \) from both sides:

$$ 30 = 5x - 5 $$

Add \( 5 \) to both sides:

$$ 35 = 5x $$

Divide both sides by \( 5 \):

$$ x = 7 $$

Answer:

\( x = 7 \)