Certainly! Let's solve this step by step.
Given:
- The square of the hypotenuse of an isosceles right-angled triangle is 1250 cm².
First, let us denote the length of the hypotenuse by [tex]\( c \)[/tex].
1. Calculate the hypotenuse length:
[tex]\[
c^2 = 1250 \text{ cm}^2
\][/tex]
To find [tex]\( c \)[/tex], we take the square root of both sides:
[tex]\[
c = \sqrt{1250} \approx 35.355 \text{ cm}
\][/tex]
2. Understand the isosceles right-angled triangle properties:
In an isosceles right-angled triangle, the two legs (let's denote the length of each leg by [tex]\( a \)[/tex]) are equal.
Thus, we can use the Pythagorean theorem:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
Simplifying this, we get:
[tex]\[
2a^2 = c^2
\][/tex]
3. Solve for the length of each leg [tex]\( a \)[/tex]:
Substitute [tex]\( c^2 = 1250 \)[/tex] into the equation:
[tex]\[
2a^2 = 1250
\][/tex]
Divide both sides by 2:
[tex]\[
a^2 = 625
\][/tex]
Take the square root of both sides to find [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{625} = 25 \text{ cm}
\][/tex]
Therefore, the length of each side of the isosceles right-angled triangle is 25 cm, and the length of the hypotenuse is approximately 35.355 cm.
