(i) The lengths of the sides of a triangle are 9, 40, 41. State
whether the triangle is right angled triangle.
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★ How to do :-

Here, we are given with three sides of a triangle. We are asked to find out that whether it’s a right angled triangle. To prove that we need some other concepts which are applied to the given numbers. Here, we are going to use the square numbers, in which we are going to multiply the number two times. The other main concept used here is the Pythagoras theorem. This property is only applied for rights angled triangles. To prove that it’s a right angled triangle, this is only the way to find. So, let’s solve!!

[tex]\:[/tex]

➤ Solution :-

We know that, in a right angled triangle always the hypotnous side is the greater than other two sides. So,

[tex]{\tt \leadsto \underline{\boxed{\tt {AB}^{2} + {BC}^{2} = {AC}^{2}}}}[/tex]

Substitute the values.

[tex]{\tt \leadsto {9}^{2} + {40}^{2} = {41}^{2}}[/tex]

Now, find the value of square numbers of them.

[tex]{\tt \leadsto 81 + 1600 = 1681}[/tex]

Now, add the numbers on LHS.

[tex]{\tt \leadsto 1681 = 1681}[/tex]

Here, we can observe that both the sides are equal

So,

[tex]{\tt \leadsto LHS = RHS}[/tex]

[tex]\Huge\therefore[/tex] The sides 9, 40 and 41 can form a right angled triangle.

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More to know :–

• The property used here is Pythagoras theorem which was proved by a scientist called Pythagoras.
• This rule says that the sum of square of the other two arms in a rights angled triangle should always equal to the hypotnous.
• The side hypotnous is always the biggest side in right angled triangle. It lies in the opposite side of the angle forming 90°.