Question

In the figure find $PM$.

A. 2 cm
B. 3 cm
C. 4 cm
D. 5 cm

Answer 1

Hint: We are given a figure such that we can see two different triangles. To find the value of PM we have to first show that the two triangles are congruent. Such that, if the value of PL is found, that will be the value for PM.

Complete step by step answer:
Given from the figure let’s consider \[\vartriangle LOP\& \vartriangle MOP\]
We can observe that,
\[\angle L = \angle M\]….both are right angles
\[\Rightarrow \angle O = \angle O\]…..common angle
Thus by AA test \[\vartriangle LOP \cong \vartriangle MOP\]
So we can say that \[LP = MP\]….elements of congruent triangles
Now we will find the value of LP.
From \[\vartriangle LOP\],
Using Pythagoras theorem,
\[LP = \sqrt {O{P^2} - O{L^2}} \]
Putting the values from the figure,
\[LP = \sqrt {{5^2} - {3^2}} \]
Taking the squares,
\[LP = \sqrt {25 - 9} \]
\[\Rightarrow LP = \sqrt {16} \]
\[\therefore LP = 4\,cm\]
Thus value of LP=MP=4 cm

Thus option C is correct.

Note: The value of PM is found with the help of the other triangle. There is no such need to find the value of all the other sides of \[\vartriangle MOP\]. Also note that, O is the common angle for both the triangles. Like AA property we have SSS, SAS and ASA tests to prove that triangles are congruent in general.