What is the perimeter of an isosceles right triangle that has two legs that measure 5 cm?

A) 12.5 cm
B) 15 cm
C) [tex]\(10 + 5\sqrt{2}\)[/tex] cm
D) [tex]\(15 + 5\sqrt{3}\)[/tex] cm



Answer :

Certainly! Let's solve for the perimeter of an isosceles right triangle where each leg measures 5 cm.

1. Identify triangle properties:
- In an isosceles right triangle, the two legs are equal.
- Let's denote the length of each leg as [tex]\( a \)[/tex].

2. Given values:
- Each leg length [tex]\( a = 5 \)[/tex] cm.

3. Calculate the hypotenuse [tex]\( c \)[/tex]:
- According to the Pythagorean theorem:
[tex]\[ c = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \][/tex]
- Substitute the given value [tex]\( a = 5 \)[/tex]:
[tex]\[ c = 5\sqrt{2} \approx 7.0710678118654755 \text{ cm} \][/tex]

4. Calculate the perimeter [tex]\( P \)[/tex]:
- The perimeter of the triangle is the sum of the lengths of its three sides:
[tex]\[ P = a + a + c = 2a + c \][/tex]
- Substitute [tex]\( a = 5 \)[/tex] and [tex]\( c = 5\sqrt{2} \)[/tex]:
[tex]\[ P = 2(5) + 5\sqrt{2} = 10 + 5\sqrt{2} \approx 17.071067811865476 \text{ cm} \][/tex]

The perimeter of the isosceles right triangle is [tex]\( 10 + 5\sqrt{2} \)[/tex] cm.

So, the correct answer is:
C) [tex]\( 10 + 5\sqrt{2} \)[/tex] cm.