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

In rhombus \( A B C D \), the diagonals bisect each other at right angles. This means that \( E \) is the midpoint of both diagonals \( \overline{A C} \) and \( \overline{B D} \). Given \( A E = 5 \) and \( B E = 12 \), we can find the lengths of the sides of the rhombus using the Pythagorean theorem: Let the sides of the rhombus be represented as \( AB = s \). Since diagonals bisect each other, triangle \( ABE \) is a right triangle where \( A E \) and \( B E \) are the legs. Thus, we can use the formula: \[ AB = \sqrt{(AE)^2 + (BE)^2} = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13. \] Therefore, the length of \( \overline{A B} \) is \( 13 \). The correct answer is option 3) 13.