Question

From the figure PT is an altitude of the triangle PQR in which \[PQ = 25cm\], \[PR = 17cm\] and \[PT = 15cm\]. If \[QR = xcm\]. Calculate $x$.

Answer 1

Hint:
Here, we have given a triangle PQR in which the PT is an altitude and we have given two sides and have to calculate the third side which is perpendicular to the altitude. So, firstly we will apply the Pythagoras theorem on the triangle PTQ and PTR in which we will get the value of QT and TR. The sum of QT and TR is equal to the QR.

Complete step by step solution:
In triangle PQR the PT is an altitude.
 \[PQ = 25cm\], \[PT = 15cm\], \[PR = 17cm\] and \[QR = x cm\].
As shown in figure triangle PTQ and triangle are right angled triangles.
Now apply Pythagoras theorem on triangle PTQ.
 \[
   \Rightarrow P{T^2} + Q{T^2} = P{Q^2} \\
   \Rightarrow {(15)^2} + Q{T^2} = {(25)^2} \\
   \Rightarrow Q{T^2} = {(25)^2} - {(15)^2} \\
   \Rightarrow Q{T^2} = 625 - 225 \\
   \Rightarrow Q{T^2} = 400 \\
   \Rightarrow QT = 20
 \]
Now apply Pythagoras theorem on triangle PTR
 $
   \Rightarrow P{T^2} + T{R^2} = P{R^2} \\
   \Rightarrow {(15)^2} + T{R^2} = {(17)^2} \\
   \Rightarrow T{R^2} = 289 - 225 \\
   \Rightarrow T{R^2} = 64 \\
   \Rightarrow TR = 8
 $
Now the value of QR(x)
 $
   = PT + TR \\
   = 20 + 8 \\
   = 28cm
 $

Therefore, the value of \[QR = 28cm\].

Note:
Here we have used the Pythagoras theorem because the PT is an altitude of the PQR.
Pythagoras theorem states that in the right angled triangle the square of length of hypotenuse is equal to the sum of squares of length of the other two sides. i.e. $A{B^2} = A{C^2} + C{B^2}$