Question

explore 1
pythagorean theorem
reflect
how can you use the pythagorean theorem to prove whether a triangle is a right triangle?

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem states that for a right - triangle with legs of length \(a\) and \(b\), and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\), where \(c\) is the longest side (opposite the right angle).

Step2: Apply the Theorem to a Triangle

To determine if a triangle with side lengths \(x\), \(y\), and \(z\) (where \(z\) is the longest side) is a right - triangle, we calculate \(x^{2}+y^{2}\) and \(z^{2}\).

• If \(x^{2}+y^{2}=z^{2}\), then by the converse of the Pythagorean Theorem, the triangle is a right - triangle. The converse of the Pythagorean Theorem states that if the sum of the squares of the two shorter sides of a triangle is equal to the square of the longest side, then the triangle is a right - triangle.
• If \(x^{2}+y^{2}

eq z^{2}\), then the triangle is not a right - triangle.

Answer:

To prove if a triangle is a right - triangle using the Pythagorean Theorem, follow these steps:

1. Identify the lengths of the three sides of the triangle. Let the lengths be \(a\), \(b\), and \(c\), where \(c\) is the longest side.
2. Calculate \(a^{2}+b^{2}\) and \(c^{2}\) using the formulas \(a^{2}=a\times a\), \(b^{2}=b\times b\), and \(c^{2}=c\times c\).
3. Compare the two results:

• If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right - triangle (by the converse of the Pythagorean Theorem).
• If \(a^{2}+b^{2}

eq c^{2}\), then the triangle is not a right - triangle.