In rhombus \( A B C D \), the diagonals \( \overline{A C} \) and \( \overline{B D} \) intersect at \( E \). If \( A E=5 \) and \( B E=12 \), what is the length of \( \overline{A B} \) ? 1) 7 2) 10 3) 13 4) 17

To find the length of \( \overline{A B} \) in rhombus \( ABCD \), we first note that the diagonals intersect at right angles and bisect each other. Given \( AE = 5 \) and \( BE = 12 \), we can determine the lengths of the diagonals: - The full length of diagonal \( AC \) is \( AE + EC = 5 + 5 = 10 \). - The full length of diagonal \( BD \) is \( BE + ED = 12 + 12 = 24 \). Now we can use the Pythagorean theorem to find \( AB \): \[ AB^2 = AE^2 + BE^2 = 5^2 + 12^2 = 25 + 144 = 169 \] Taking the square root gives: \[ AB = \sqrt{169} = 13 \] Thus, the length of \( \overline{A B} \) is \( 13 \). Therefore, the answer is \( \boxed{13} \).