Question

use pythagorean theorem to prove a triangle is a right triangle or not. are these triangles right triangles or not? triangle with vertices p, d, e: pd=5, de=12, pe=13; triangle with sides 6, 9, 11

Explanation:

Step1: Analyze Triangle PDE

Identify the sides: \( a = 5 \), \( b = 12 \), \( c = 13 \) (hypotenuse candidate).
Check Pythagorean Theorem: \( a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169 \), \( c^2 = 13^2 = 169 \).
Since \( 5^2 + 12^2 = 13^2 \), triangle PDE is a right triangle.

Step2: Analyze the Second Triangle

Identify the sides: \( a = 6 \), \( b = 9 \), \( c = 11 \) (hypotenuse candidate).
Check Pythagorean Theorem: \( a^2 + b^2 = 6^2 + 9^2 = 36 + 81 = 117 \), \( c^2 = 11^2 = 121 \).
Since \( 6^2 + 9^2
eq 11^2 \) (117 ≠ 121), the second triangle is not a right triangle.

Answer:

• Triangle PDE (with sides 5, 12, 13): Right Triangle.
• Triangle with sides 6, 9, 11: Not a Right Triangle.