2021 JMPSC Accuracy Problems/Problem 6

Revision as of 12:54, 11 July 2021 by Grisham (talk | contribs)

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 $28+28 = \boxed{56}$

~Grisham

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