Triangle GFY is right angle triangle because it satisfies both methods (distance and slope) for right triangle.
How to prove that a triangle is a right angle triangle.
Method 1: Distance Criteria
Calculate Distances:
GF: √{(4 - (-2))² + ((-1) - (-3))³}
= √{6² + 2²} = {36 + 4}
= √40 = 2√10
GY : √{(1 - (-2))² + (8 - (-3))²}
= √{3² + 11²} = √{9 + 121} = √130
FY: √(1 - 4)² + (8 - (-1))²}
= √{(-3)² + 9²} = √{9 + 81}
= √90 = 3√10
Pythagorean Theorem:
GY² = FY² + GF²
(√130)² = (3√10)² + (2√10)²
130 = 90 + 40
130 = 130
Since the equation is satisfied triangle GFY is a right triangle based on the distance criteria.
Method 2: Slope Criteria
Calculate Slopes:
mGF = (-1 - (-3)/(4 - (-2))
= 2/6 = 1/3
mGY = (8 - (-3))/(1 - (-2))
= 11/3
mFY = (8 - (-1))/(1 - 4)
= 9/-3 = -3
Product of Slopes:
mGF * mGY = 1/3*11/3 = 11/9
mGY *mFY = 11/3*-3 = -11
mGF * mFY = 1/3 *-3 = -1
Since mGF *mFY = -1 triangle GFY satisfies the slope criteria, and it is a right triangle based on the slope criteria.