A side of an isosceles right angled triangle is x. Find its hypotenuse.
Chapter 2 – Pythagoras Theorem- Text Book Solution
Problem Set 2 | Q 2.5 | Page 44
A side of an isosceles right angled triangle is x. Find its hypotenuse.
It is given that, a side of an isosceles right-angled triangle is x.
Then, the other side of the triangle is also x.
According to Pythagoras theorem.
\[\left( \text{Hypotenuse} \right)^2 = x^2 + x ^2 \]
\[ = 2 x^2 \]
\[ \therefore \text{Hypotenuse} = x\sqrt{2}\]
Hence, its hypotenuse is x\[\sqrt{2}\]
Explanation:-
In an isosceles right-angled triangle, the two legs (or sides) that form the right angle are congruent (i.e., they have the same length), so we have:
leg1 = leg2 = x
According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Let’s use this theorem to find the length of the hypotenuse of the isosceles right-angled triangle.
So, we have:
hypotenuse^2 = leg1^2 + leg2^2
Substituting leg1 = leg2 = x, we get:
hypotenuse^2 = x^2 + x^2
hypotenuse^2 = 2x^2
Taking the square root of both sides, we get:
hypotenuse = sqrt(2x^2)
We can simplify this expression using the property of square roots that says sqrt(a^2) = |a|. So, we have:
hypotenuse = |x| * sqrt(2)
Note that we take the absolute value of x because x could be positive or negative, but the length of a side cannot be negative.
Therefore, the length of the hypotenuse of the isosceles right-angled triangle is |x| * sqrt(2).
Chapter 2 – Pythagoras Theorem- Text Book Solution
Problem Set 2 | Q 2.5 | Page 44