Question

Consider an isosceles right angle with hypotenuse length of 10. Exactly how long are the legs?

Answer 1

Hint: An isosceles right-angle triangle is a combination of an isosceles triangle and a right-angled triangle, in this triangle, one of the angles is of measure 90 degrees and two sides of the triangle are equal. Using Pythagora's theorem, we can find the relation between the three sides of the triangle and thus get an equation to find out the unknown quantities.

Complete step-by-step answer:
Let the base and the height of the isosceles right-angle triangle be of the length x units and we are given that the length of the hypotenuse is 10.
By Pythagoras theorem, we get –
 $
  {x^2} + {x^2} = {(10)^2} \\
   \Rightarrow 2{x^2} = 100 \\
   \Rightarrow {x^2} = \dfrac{{100}}{2} \\
   \Rightarrow x = \pm \sqrt {50} \\
   \Rightarrow x = \pm 5\sqrt 2 \;
  $
As length cannot be negative, the negative value is rejected.
Hence, the legs of this triangle are $ 5\sqrt 2 $ long.
So, the correct answer is “ $ 5\sqrt 2 $ ”.

Note: The base and the height are referred to as the legs of the triangle in the given question. From the Pythagoras theorem, we see that the hypotenuse is the sum of the squares of the base and the perpendicular, so it is greater than both the sides and thus can’t be equal to any of them that is why base and height are taken to be equal.