Question

in circle a, ∠bae ≅ ∠dae. what is the length of \\(\overline{be}\\)? 34 units 17 units 27 units 14 units

Explanation:

Step1: Identify Congruent Triangles

Since \( \angle BAE \cong \angle DAE \), \( AB = AD \) (radii of the circle), and \( AE \) is common, \( \triangle BAE \cong \triangle DAE \) by SAS. Thus, \( BE = DE \).

Step2: Set Up Equation for \( BE \) and \( DE \)

\( BE = 3x - 24 \) and \( DE = x + 10 \). Since \( BE = DE \), set \( 3x - 24 = x + 10 \).

Step3: Solve for \( x \)

Subtract \( x \) from both sides: \( 2x - 24 = 10 \). Add 24 to both sides: \( 2x = 34 \). Divide by 2: \( x = 17 \).

Step4: Calculate \( BE \)

Substitute \( x = 17 \) into \( BE = 3x - 24 \): \( BE = 3(17) - 24 = 51 - 24 = 27 \).

Answer:

27 units