# How do you find the hypotenuse of an isosceles right triangle?

How do you find the hypotenuse of an isosceles right triangle?

296.1k+ views

Hint: In this question, we have to find out the required expression from the given particulars.
We need to first consider what an isosceles right triangle is. For that we need to know the definition of an isosceles triangle right triangle
We need to apply Pythagoras theorem on this triangle, so we can find out the required solution.
Complete step-by-step solution:
We need to find the hypotenuse of an isosceles right triangle.
In geometry, an isosceles triangle is a triangle that has two sides of equal length.
In geometry, a right angled triangle is a triangle that has one right angle.

Let , ABC is an isosceles right triangle with $\angle ABC = 90^\circ$.
Since it is also an isosceles triangle, let AB = BC.
Also as it is a right angle triangle we can apply Pythagoras theorem which states,
${\left( {Hypotenuse} \right)^2} = {\left( {Height} \right)^2} + {\left( {Base} \right)^2}$
i.e.$A{C^2} = A{B^2} + B{C^2}$
i.e.$A{C^2} = A{B^2} + A{B^2}$, [since, AB = BC].
i.e.$A{C^2} = 2A{B^2}$
on taking square root on both sides we get,
i.e.$AC = \sqrt {2A{B^2}}$
on rewriting we get
i.e.$AC = \sqrt 2 AB$
Hence, the hypotenuse of an isosceles right triangle is $\sqrt 2$ $\times$ the equal side of the triangle.
Note: Right angled triangle:
In geometry, a right angled triangle is a triangle that has one right angle.
Pythagoras theorem states that,
If ABC is a right angled triangle then,
${\left( {Hypotenuse} \right)^2} = {\left( {Height} \right)^2} + {\left( {Base} \right)^2}$
$A{C^2} = A{B^2} + B{C^2}$
Isosceles triangle:
In geometry, an isosceles triangle is a triangle that has two sides of equal length.
An isosceles triangles definition states it as a polygon that consists of two equal sides, two equal angles, three edges, three vertices and the sum of internal angles of a triangle equal to $180^\circ$.

We need to first consider what an isosceles right triangle is. For that we need to know the definition of an isosceles triangle right triangle

We need to apply Pythagoras theorem on this triangle, so we can find out the required solution.

Complete step-by-step solution:

We need to find the hypotenuse of an isosceles right triangle.

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

In geometry, a right angled triangle is a triangle that has one right angle.

Let , ABC is an isosceles right triangle with $\angle ABC = 90^\circ$.

Since it is also an isosceles triangle, let AB = BC.

Also as it is a right angle triangle we can apply Pythagoras theorem which states,

${\left( {Hypotenuse} \right)^2} = {\left( {Height} \right)^2} + {\left( {Base} \right)^2}$

i.e.$A{C^2} = A{B^2} + B{C^2}$

i.e.$A{C^2} = A{B^2} + A{B^2}$, [since, AB = BC].

i.e.$A{C^2} = 2A{B^2}$

on taking square root on both sides we get,

i.e.$AC = \sqrt {2A{B^2}}$

on rewriting we get

i.e.$AC = \sqrt 2 AB$

Hence, the hypotenuse of an isosceles right triangle is $\sqrt 2$ $\times$ the equal side of the triangle.

Note: Right angled triangle:

In geometry, a right angled triangle is a triangle that has one right angle.

Pythagoras theorem states that,

If ABC is a right angled triangle then,

${\left( {Hypotenuse} \right)^2} = {\left( {Height} \right)^2} + {\left( {Base} \right)^2}$

$A{C^2} = A{B^2} + B{C^2}$

Isosceles triangle:

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

An isosceles triangles definition states it as a polygon that consists of two equal sides, two equal angles, three edges, three vertices and the sum of internal angles of a triangle equal to $180^\circ$.

Last updated date: 23rd Aug 2023

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