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Question :
Proof that if (-2,5),(1,3),(5,9) is make a right triangle.
clue : materi persamaan garis

Answer 1

Let A = (-2,5), B = (1,3), C = (5,9)

the length of three lines respectively are
AB = √(3 - 5)²+(1-(-2))² = √(4+9) = √13
AC = √(9 - 5)²+(5 - (-2))² = √(16+49) = √65
BC = √(9 - 3)² + (5 - 1)² = √(36+16) = √52

then use the pythagorena theorem, it should be satisfies
AB² + BC² = AC²
√13² + √52² = √65²
13 + 52 = 65
(it is a right triangle)

proved

Answer 2

By using phitaghoras

distance (-2,5) to (1,3) and (1,3) to (5,9) as same as distance (-2,5) to (5,9)
Prove:
[√((3 - 5)² + (1 + 2)²) ]² + [√((9 - 3)² + (5 - 1)²)]² = [√((9 - 5)² + (5 + 2)²)]²
[√(4 + 9) ]² + [√(36 + 16)]² = [√(16 + 49)]²
13 + 52 = 65
65 = 65