Question

Which identity is the result of using the Pythagorean theorem to show that a triangle with side lengths x, 1, 2x, and (x² + 1) is a right triangle?

A. (x² + 1 = 2x²)
B. (2x² = x² + 1)
C. (x² + 1 = 1 + 2x)
D. (2x² = x² + 2)

Answer 1

The result after applying the Pythagorean theorem is (2x²= x² + 1), which corresponds to option B. This identity validates the triangle as a right triangle when satisfied by the given side lengths x, 1, 2x, and (x² + 1).

Step-by-step explanation:

The Pythagorean theorem states that for 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, the sides of the triangle are x, 1, 2x, and (x² + 1). Applying the theorem:

(x² + 1²= (2x)²)

(x² + 1 = 4x²)

(1 = 4x² - x²)

(1 = 3x²)

(x² =
`(1)/(3)\)`

2x² =
`(2)/(3)\)`

Upon rearranging the equation, we get:

[2x² - x² = 1]

[x² = 1]

[2x² = x² + 1]

Thus, the identity resulting from using the Pythagorean theorem on the given triangle is (2x² = x² + 1), which is option B.

This outcome verifies that the triangle satisfies the conditions for being a right triangle. Therefore, when the values of x satisfy the equation (2x² = x² + 1), the triangle with side lengths x, 1, 2x, and (x² + 1) forms a right triangle, validating the Pythagorean theorem.