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

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1. In a rhombus, the diagonals are perpendicular bisectors of each other. Therefore, triangle $$ AEB $$ is a right triangle with the right angle at $$ E $$.

2. The lengths of $$ AE $$ and $$ BE $$ are given as 5 and 12 respectively. Thus, in triangle $$ AEB $$, these are the lengths of the legs.

3. By the Pythagorean theorem, the side length $$ AB $$ is given by
   $$
   AB = \sqrt{AE^2 + BE^2}.
   $$
   
4. Substitute the given values:
   $$
   AB = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169}.
   $$
   
5. Simplify the square root:
   $$
   AB = 13.
   $$

The length of side $$ \overline{AB} $$ is $$ 13 $$.
The length of $$ \overline{AB} $$ is 13.

In rhombus , the diagonals and intersect at . If and , what is the length of ?

7

10

13

17

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In rhombus , the diagonals and intersect at . If and , what is the length of ? 7 10 13 17