Naomi takes a 52-inch by 39-inch rectangle of fabric and cuts from one corner of the piece of fabric to the diagonally opposite corner. Now Naomi has two equally sized triangles of fabric. What is the perimeter of each triangle?

Use the Pythagorean theorem to find the length of the hypotenuse.



Answer :

Certainly! Let's walk through the solution step-by-step:

### Step 1: Understand the Problem
Naomi has a rectangle with dimensions 52 inches by 39 inches. She cuts the rectangle diagonally from one corner to the opposite corner, forming two right-angled triangles of fabric. We need to determine the perimeter of each of these triangles.

### Step 2: Determine the Hypotenuse
In a right-angled triangle, the hypotenuse can be found using the Pythagorean theorem. For a right-angled triangle with legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the hypotenuse [tex]\(c\)[/tex] can be calculated as:
[tex]\[ c = \sqrt{a^2 + b^2} \][/tex]

Here, the legs of the triangle are 52 inches and 39 inches.

### Step 3: Calculate the Hypotenuse
Using the given dimensions:
[tex]\[ a = 52 \][/tex]
[tex]\[ b = 39 \][/tex]

The hypotenuse [tex]\(c\)[/tex] is:
[tex]\[ c = \sqrt{52^2 + 39^2} \][/tex]
[tex]\[ c = \sqrt{2704 + 1521} \][/tex]
[tex]\[ c = \sqrt{4225} \][/tex]
[tex]\[ c = 65.0 \][/tex]

### Step 4: Calculate the Perimeter of One Triangle
The perimeter of a triangle is the sum of the lengths of its three sides. For each of the right-angled triangles obtained by cutting the rectangle diagonally, the sides are 52 inches, 39 inches, and the hypotenuse 65 inches.

Thus, the perimeter [tex]\(P\)[/tex] of one triangle is:
[tex]\[ P = 52 + 39 + 65 \][/tex]
[tex]\[ P = 156.0 \][/tex]

### Conclusion
The hypotenuse of each triangle is 65 inches, and the perimeter of each triangle is 156 inches.

Therefore, the perimeter of each triangle is 156 inches.