Question

Which of the following triplets is not the right triangle?
(a) 7, 24, 25
(b) 60, 80, 90
(c) 5, 12, 13
(d) 9, 40, 41

Answer 1

Hint: First, let us draw a diagram of a right-angled triangle for the reference. Let use the method of trial and error method to check whether which one of the options is not a right-angled triangle with the help of the Pythagoras theorem, ${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}}$. Substitute the values in the formula and check if L.H.S is equal to R.H.S, the one which is not equal is the required answer.

Complete step-by-step solution
We have 4 options out of which three of them are the sides of the right-angled triangle, here we have to find the one which is not a right-angled triangle.
First, let us consider a triangle ABC which is a triangle that has three sides AB, BC, and AC, and let us draw a right-angled triangle for reference.

Let us check for the first option,
(a) 7, 24, 25:
We need to check if these three sides make a right-angled triangle.
In a right-angled triangle, the hypotenuse is the longest side. Hence, 25 is the hypotenuse and 7 and 24 are the remaining two sides.
From Pythagoras theorem, we know
${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}}$
So, if the right-hand side equals the left-hand side, we can say that the respective triangle is a right-angled triangle.
 $\begin{align}
  & \text{L}\text{.H}\text{.S = }{{\left( \text{Hypotenuse} \right)}^{2}} \\
 & ={{\left( 25 \right)}^{2}} \\
 & =625
\end{align}$
$\begin{align}
  & \text{R}\text{.H}\text{.S = }{{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}} \\
 & ={{\left( 7 \right)}^{2}}+{{\left( 24 \right)}^{2}} \\
 & =49+576 \\
 & =625
\end{align}$
Here, L.H.S = R.H.S, therefore we can say that the sides are of right-angled triangles.
(b) 60, 80, 90
Similarly, let us check for these sides of the triangle.
Hypotenuse = 90, side one = 60, side two = 80
From Pythagoras theorem, we know
${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}}$
$\begin{align}
  & \text{L}\text{.H}\text{.S =}{{\left( \text{Hypotenuse} \right)}^{2}} \\
 & ={{\left( 90 \right)}^{2}} \\
 & =8100
\end{align}$
$\begin{align}
  & \text{R}\text{.H}\text{.S = }{{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}} \\
 & ={{\left( 60 \right)}^{2}}+{{\left( 80 \right)}^{2}} \\
 & =3600+6400 \\
 & =10000
\end{align}$
Here,$\text{L}\text{.H}\text{.S}\ne \text{R}\text{.H}\text{.S}$, therefore these three sides do not make a right-angled triangle.
For option (c):
Hypotenuse = 13, side one = 12, side two = 5
From Pythagoras theorem, we get
${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}}$
$\begin{align}
  & \text{L}\text{.H}\text{.S =}{{\left( \text{Hypotenuse} \right)}^{2}} \\
 & ={{\left( 13 \right)}^{2}} \\
 & =169
\end{align}$
\[\begin{align}
  & \text{R}\text{.H}\text{.S = }{{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}} \\
 & ={{\left( 12 \right)}^{2}}+{{\left( 5 \right)}^{2}} \\
 & =144+25 \\
 & =169
\end{align}\]
Here, L.H.S = R.H.S, therefore we can say that the sides are of right-angled triangles.
For option (d):
Hypotenuse = 41, side one = 9, side two = 40
From Pythagoras theorem, we get
${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}}$
$\begin{align}
  & \text{L}\text{.H}\text{.S =}{{\left( \text{Hypotenuse} \right)}^{2}} \\
 & ={{\left( 41 \right)}^{2}} \\
 & =1681
\end{align}$
\[\begin{align}
  & \text{R}\text{.H}\text{.S = }{{\left( \text{side one} \right)}^{2}}+{{\left( \text{side two} \right)}^{2}} \\
 & ={{\left( 9 \right)}^{2}}+{{\left( 40 \right)}^{2}} \\
 & =81+1600 \\
 & =1681
\end{align}\]
Here, L.H.S = R.H.S, therefore we can say that the sides are of right-angled triangles.
Hence, only option (b) is the triplet which is not a right-angled triangle.

Note: In a right-angled triangle, there are mainly two types of a right-angled triangle. They are classified with respect to the angles in the triangle. The two types are, (30 – 60 – 90) which indicates the three angles in the right-angled triangle, the next is (45 – 45 – 90) which is an isosceles right-angled triangle since two of its angles are congruent which indicates that two opposite sides of those equal angles will also be congruent.