INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

June 12, 2020

Finding side of Triangle | PRMO-2014 | Problem 15

Try this beautiful problem from PRMO, 2014 based on Finding side of Triangle.

Finding side of Triangle | PRMO | Problem 15


Let XOY be a triangle with angle XOY=90 degrees. Let M and N be the midpoints of the legs OX and OY, respectively. Suppose that XN=19 and YM=22. what is XY?

  • \(28\)
  • \(26\)
  • \(30\)

Key Concepts


Geometry

Triangle

Pythagoras

Check the Answer


Answer:\(26\)

PRMO-2014, Problem 15

Pre College Mathematics

Try with Hints


Finding side of Triangle - figure

Given that \(\angle XOY=90^{\circ}\) .Let M and N be the midpoints of the legs OX and OY, respectively, and that XN=19 and YM=22. Now \(\triangle XON\) & \(\triangle MOY\) are Right angle Triangle. Use Pythagoras theorem .......

Can you now finish the problem ..........

Triangle problem

Let \(XM=MO=p\) and \(ON=NY=q\).Now using Pythagoras theorm on \(\triangle XON\) & \(\triangle MOY\) we have...

\(OX^2 +ON^2=XN^2\) \(\Rightarrow 4p^2 +q^2=19^2\) \(\Rightarrow 4p^2 +q^2=361\)...........(1) and \(OM^2 +OY^2=MY^2\) \(\Rightarrow p^2 +4q^2=22^2\) \(\Rightarrow p^2 +4q^2=484\)......(2)

Finding side of Triangle

Now Adding (1)+(2)=\((4p^2 +q^2=361)\)+\((p^2 +4q^2=484\) \(\Rightarrow 5(p^2+q^2)=845\) \(\Rightarrow (p^2+q^2)=169\) \(\Rightarrow 4(p^2+q^2)=676\) \(\Rightarrow (OX)^2+(OY)^2=(26)^2\) \(\Rightarrow (XY)^2=(26)^2\) \(\Rightarrow XY=26\).

Subscribe to Cheenta at Youtube


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter