Question

the hypotenuse of an isosceles right triangle is 21 centimeters longer than either of its legs. find the exact length of each side. (hint: an isosceles right triangle is a right triangle whose legs are the same length )
the length of one leg is \\(\square\\) \\(\boldsymbol{\
abla}\\) the length of the other leg is \\(\square\\) \\(\boldsymbol{\
abla}\\) and the length of the hypotenuse is \\(\square\\) \\(\boldsymbol{\
abla}\\) (simplify your answers, inclu use integers or fractions for any numbers in the expressions )
cm,
cu cm,
sq cm,

Explanation:

Step1: Define variable for leg length

Let leg length = $x$ cm. Hypotenuse = $x + 21$ cm.

Step2: Apply Pythagorean theorem

$x^2 + x^2 = (x + 21)^2$ → $2x^2 = x^2 + 42x + 441$

Step3: Simplify quadratic equation

$x^2 - 42x - 441 = 0$

Step4: Solve using quadratic formula

$x = \frac{42 \pm \sqrt{42^2 + 4 \times 441}}{2} = \frac{42 \pm \sqrt{1764 + 1764}}{2} = \frac{42 \pm 42\sqrt{2}}{2} = 21(1 + \sqrt{2})$ (discard negative)

Step5: Find hypotenuse length

Hypotenuse = $21(1 + \sqrt{2}) + 21 = 21(2 + \sqrt{2})$

Answer:

One leg: $21(1 + \sqrt{2})$ cm
Other leg: $21(1 + \sqrt{2})$ cm
Hypotenuse: $21(2 + \sqrt{2})$ cm