Question

The hypotenuse of an isosceles right triangle is \[12\] inches. What is the length of each leg?

Answer 1

Hint: We have to find the length of each leg of an isosceles right triangle whose hypotenuse is given as \[12\] inches. As this is an isosceles right triangle so length of each leg will be equal. We will assume the length of each leg to be \[x\]. Then in a right-angled triangle we can apply Pythagoras theorem, so we will apply the Pythagoras theorem and then simplify it to find the result.

Complete step by step answer:

As we know that in an isosceles right-angle triangle two legs of the right angle are equal in length. Consider a right angled \[\vartriangle ABC\] in which \[AB\] and \[BC\] are the legs of the triangle. Since it is an isosceles right triangle, therefore, \[AB\] and \[BC\] are equal.Let, \[AB = BC = x\].Now, \[\vartriangle ABC\] is a right-angled triangle. So, we can apply Pythagoras theorem.According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle. So, we can write
\[ \Rightarrow A{C^2} = A{B^2} + B{C^2}\]

Putting the values, we get
\[ \Rightarrow {\left( {12} \right)^2} = {\left( x \right)^2} + {\left( x \right)^2}\]
\[ \Rightarrow 144 = {x^2} + {x^2}\]
On simplifying, we get
\[ \Rightarrow 144 = 2{x^2}\]
On rewriting, we get
\[ \Rightarrow 2{x^2} = 144\]

Dividing both the sides by \[2\], we get
\[ \Rightarrow {x^2} = \dfrac{{144}}{2}\]
On calculating, we get
\[ \Rightarrow {x^2} = 72\]
Taking square root both sides, we get
\[ \Rightarrow x = \pm \sqrt {72} \]
But length can’t be negative. So, \[x \ne - \sqrt {72} \].
Therefore, we get
\[ \Rightarrow x = \sqrt {72} \]
\[ \therefore x = 6\sqrt 2 \]

Therefore, the length of each leg is \[6\sqrt 2 \] inches.

Note: The two perpendicular sides of a right triangle are called the legs. In an isosceles right triangle, the two legs are of equal length. So, the corresponding angles are also congruent. The corresponding angles are \[45\] degrees each. For any right-angle triangle, Pythagoras theorem is the most important formula.