Question

in circle a, $\angle bae \cong \angle dae$. what is the value of $x$? $\bigcirc$ 14 $\bigcirc$ 17 $\bigcirc$ 27 $\bigcirc$ 34 (with a circle diagram labeled e, b, a, d, with segments eb labeled $3x - 24$, ed labeled $x + 10$, and angles at a marked as congruent)

Explanation:

Step1: Identify congruent chords

Since \( \angle BAE \cong \angle DAE \), the chords \( BE \) and \( DE \) are congruent (In a circle, congruent central angles subtend congruent chords). So, \( BE = DE \).

Step2: Set up the equation

Given \( BE = 3x - 24 \) and \( DE = x + 10 \), we set them equal: \( 3x - 24 = x + 10 \).

Step3: Solve for x

Subtract \( x \) from both sides: \( 3x - x - 24 = 10 \) → \( 2x - 24 = 10 \).
Add 24 to both sides: \( 2x = 10 + 24 \) → \( 2x = 34 \).
Divide by 2: \( x = \frac{34}{2} = 17 \).

Answer:

17