Difference between revisions of "2021 JMPSC Sprint Problems/Problem 16"

 
Line 10: Line 10:
  
 
~Mathdreams
 
~Mathdreams
 +
 +
== Solution 2 ==
 +
<cmath>[ACD] = \frac{24 \cdot 20}{2}=240</cmath>
 +
<cmath>[ABC] = \frac{12 \cdot 16}{2}=96</cmath>
 +
Therefore, <math>[ABCD] = 240-96=144</math>
 +
 +
- kante314 -
  
 
==See also==
 
==See also==

Latest revision as of 10:39, 12 July 2021

Problem

$ABCD$ is a concave quadrilateral with $AB = 12$, $BC = 16$, $AD = CD = 26$, and $\angle ABC=90^\circ$. Find the area of $ABCD$.

Sprint16.jpg

Solution

Notice that $[ABCD] = [ADC] - [ABC]$ and $AC = \sqrt{12^2 + 16^2} = 20$ by the Pythagorean Thereom. We then have that the area of triangle of $ADC$ is $\frac{20 \cdot \sqrt{26^2 - 10^2}}{2} = 240$, and the area of triangle $ABC$ is $\frac{12 \cdot 16}{2} = 96$, so the area of quadrilateral $ABCD$ is $240 - 96 = 144$.

~Mathdreams

Solution 2

\[[ACD] = \frac{24 \cdot 20}{2}=240\] \[[ABC] = \frac{12 \cdot 16}{2}=96\] Therefore, $[ABCD] = 240-96=144$

- kante314 -

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. JMPSC.png

Invalid username
Login to AoPS