To find the perimeter of an isosceles right triangle where the length of each leg is 3 feet, follow these steps:
1. Identify the triangle type and leg lengths:
An isosceles right triangle means it has two equal legs and a right angle between them. Here, each leg is given as 3 ft.
2. Apply the Pythagorean Theorem:
To find the length of the hypotenuse (the side opposite the right angle), use the Pythagorean theorem, which states:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
For our triangle, both legs are 3 ft:
[tex]\[
3^2 + 3^2 = c^2
\][/tex]
Simplifying this:
[tex]\[
9 + 9 = c^2
\][/tex]
[tex]\[
18 = c^2
\][/tex]
3. Solve for the hypotenuse:
Take the square root of both sides to find the hypotenuse:
[tex]\[
c = \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}
\][/tex]
Thus, the hypotenuse is [tex]\( 3\sqrt{2} \)[/tex] ft.
4. Calculate the perimeter of the triangle:
The perimeter is the sum of all three sides of the triangle:
[tex]\[
\text{Perimeter} = \text{Leg}_1 + \text{Leg}_2 + \text{Hypotenuse}
\][/tex]
Substituting the known lengths:
[tex]\[
\text{Perimeter} = 3 + 3 + 3\sqrt{2} = 6 + 3\sqrt{2} \text{ ft}
\][/tex]
So, the perimeter of the given isosceles right triangle is [tex]\( 6 + 3\sqrt{2} \)[/tex] ft.
The correct answer is:
[tex]\[ \boxed{6 + 3\sqrt{2} \text{ ft}} \][/tex]
