2021 JMPSC Accuracy Problems/Problem 6

Problem

In quadrilateral $ABCD$ , diagonal $\overline{AC}$ bisects both $\angle BAD$ and $\angle BCD$ . If $AB=15$ and $BC=13$ , find the perimeter of $ABCD$ .

Solution

Notice that since $\overline{AC}$ bisects a pair of opposite angles in quadrilateral $ABCD$ , we can distinguish this quadrilateral as a kite.

$\linebreak$ With this information, we have that $\overline{AD}=\overline{AB}=15$ and $\overline{CD}=\overline{BC}=13$ .

Therefore, the perimeter is $15+15+13+13=\boxed{56}$ $\square$

$\linebreak$ ~Apple321

Solution 2

We note that triangle $ABC$ and $DAC$ are congruent due to $AA$ congruency. Therefore, $AD + DC = 28$ and the perimeter of the quadrilateral is $28+28 = \boxed{56}$

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2021 JMPSC Accuracy Problems/Problem 6

Problem

Solution

Solution 2