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2021 JMPSC Accuracy Problems/Problem 6: Difference between revisions

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~Grisham
~Grisham
==See also==
#[[2021 JMPSC Sprint Problems|Other 2021 JMPSC Sprint Problems]]
#[[2021 JMPSC Sprint Answer Key|2021 JMPSC Sprint Answer Key]]
#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
{{JMPSC Notice}}

Revision as of 16:21, 11 July 2021

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

See also

  1. Other 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Sprint Answer Key
  3. All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.

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