Question

If the sides of the triangle are 9, 20 and 32, can we form a right angled triangle?
(a) No
(b) Yes
(c) Ambiguous
(d) Not (a) and (b)

Answer 1

Hint:First, before proceeding with this, we must know that for the right-angled triangle always one side is larger than the other two sides. Then,for the right-angled triangle, we know that by using Pythagoras Theorem the sum of the square of the base and perpendicular of the triangle is equal to the square of its hypotenuse. Then, by taking the value of the longest side which is 32 as hypotenuse and 20, 9 as base and perpendicular, and then using the Pythagoras formula given by ${{h}^{2}}={{b}^{2}}+{{p}^{2}}$, we get the required condition.

Complete step by step answer:
In this question, we are supposed to find that if a triangle is right-angled triangle when its sides are 9, 20, and 32.
So, before proceeding with this, we must know that for the right-angled triangle always one side is larger than the other two sides.
Then, w can see clearly that the side of length 32 is larger than the sides of length 9 and 20.
Also, for the right-angled triangle, we know that by using Pythagoras Theorem the sum of the square of the base and perpendicular of the triangle is equal to the square of its hypotenuse.
So, to show this diagrammatically with perpendicular p, base b and hypotenuse h, we have:

So, if the given sides of the triangle follow the above condition then we can state it as a right-angled triangle.
Now, by taking the value of the longest side which is 32 as hypotenuse and 20, 9 as base and perpendicular and then using the Pythagoras formula given by:
${{h}^{2}}={{b}^{2}}+{{p}^{2}}$
Then, by substituting the values, we get:
${{32}^{2}}={{20}^{2}}+{{9}^{2}}$
Then, by solving the above expression parallel for both sides, we get:
$\begin{align}
  & 1024=400+81 \\
 & 1024\ne 481 \\
\end{align}$
So, we can clearly see that the above expression two sides are not equal.
So, the given sides of the triangle don’t follow the Pythagoras theorem which is valid for only right-angled triangle and the answer is No.
Hence, option (a) is correct.

Note:
 Now, to solve these types of questions we need to know some of the basics of triangles in which more important is the right angles triangle. Moreover, we should also check the given sides if whether it represents a valid triangle or not by the condition that the sum of the lengths of any two sides of the triangle must be greater than or equal to the length of the third side.