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Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 6"

(Created page with "==Problem== In quadrilateral <math>ABCD</math>, diagonal <math>\overline{AC}</math> bisects both <math>\angle BAD</math> and <math>\angle BCD</math>. If <math>AB=15</math> and...")
 
(→‎Solution 2)
 
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==Solution==
 
==Solution==
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asdf
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Notice that since <math>\overline{AC}</math> bisects a pair of opposite angles in quadrilateral <math>ABCD</math>, we can distinguish this quadrilateral as a kite.
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<math>\linebreak</math>
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With this information, we have that <math>\overline{AD}=\overline{AB}=15</math> and <math>\overline{CD}=\overline{BC}=13</math>.
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Therefore, the perimeter is <cmath>15+15+13+13=\boxed{56}</cmath>
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<math>\square</math>
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<math>\linebreak</math>
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~Apple321
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==Solution 2==
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We note that triangle <math>ABC</math> and <math>DAC</math> are congruent due to <math>ASA</math> congruency. Therefore, <math>AD + DC = 28</math> and the perimeter of the quadrilateral is <math>28+28 = \boxed{56}</math>
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~Grisham
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==See also==
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#[[2021 JMPSC Accuracy Problems|Other 2021 JMPSC Accuracy Problems]]
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#[[2021 JMPSC Accuracy Answer Key|2021 JMPSC Accuracy Answer Key]]
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#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
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{{JMPSC Notice}}

Latest revision as of 16:53, 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 $ASA$ congruency. Therefore, $AD + DC = 28$ and the perimeter of the quadrilateral is $28+28 = \boxed{56}$

~Grisham

See also

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

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