Converse Of Pythagorean Theorem

Explanation
The given triangle with sides of length 5, 12, and 13 units is a right triangle because it satisfies the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, 5^2 + 12^2 = 25 + 144 = 169, which is equal to 13^2. Therefore, the given triangle is a right triangle.

Explanation
A triangle with sides of lengths 7, 11, and 13 units is not a right triangle because it does not satisfy the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In this case, 7^2 + 11^2 is not equal to 13^2, so the triangle is not a right triangle.

Explanation
A triangle with sides of length 20, 21, and 28 units cannot be a right triangle because it does not satisfy the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, if we square the lengths of the sides, we get 400, 441, and 784. However, 784 is not equal to the sum of 400 and 441, so the triangle is not a right triangle.

Explanation
A triangle is classified as a right triangle if it satisfies the Pythagorean theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the lengths of the sides are 9, 12, and 15 units. By applying the Pythagorean theorem, we can see that 9^2 + 12^2 = 81 + 144 = 225, which is equal to 15^2. Therefore, the triangle with sides measuring 9, 12, and 15 units is a right triangle.

Explanation
The triangle with sides of length 12, 25, and 27 units is not a right triangle because it does not satisfy the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, if we calculate the squares of the sides, we get 144, 625, and 729. However, 144 + 625 is not equal to 729, so the triangle is not a right triangle.

Explanation
The given triangle satisfies the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, 16^2 + 30^2 = 256 + 900 = 1156, which is equal to 34^2. Therefore, the triangle with sides of length 16, 30, and 34 units is a right triangle.

Explanation
A triangle with sides of length 3, 5, and 6 units cannot be a right triangle because it does not satisfy the Pythagorean theorem. According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. However, in this case, 3^2 + 5^2 does not equal 6^2, so the triangle cannot be a right triangle.

Explanation
The given triangle with side lengths of 7, 24, and 25 units is a right triangle because it satisfies the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, 7^2 + 24^2 = 49 + 576 = 625, which is equal to 25^2. Therefore, the triangle is a right triangle.