determine whether the points (0,4) , (2, -2), and (5, -1) are the vertices of a right triangle.

Question

 are the vertices of a right triangle

Answer 1

one way to do this would be to calculate the slopes of each side; if any two slopes m and n are such that mn = -1, then it's a right triangle because that product, negative one, indicates perpendicularity of two sides.

 

another way is to calculate all three side lengths and see if they satisfy the Pythagorean Theorem.

Answer 2

Let A = (0,4), B = (2,-2), C = (5,-1)

 

Length of AB = √[(2-0)²+(-2-4)²] = √40

 

Length of AC =  √[(5-0)²+(-1-4))²] = √50

 

Length of BC = √[(5-2)²+(-1-(-2))²] = √10

 

Since (√10)² + (√40)² = (√50)², the Pythagorean Theorem holds.  

So, ΔABC is a right triangle.

 

Answer 3

(0,4), (2,-2), (5,-1)

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

d₁ = √[(-2-4)² + (2-0)²] = √[(-6)² + 2²] = √(36 + 4) = √40

d₂ = √[(-1+2)² + (5-2)²] = √[1² + 3²] = √(1 + 9) = √10

d₃ = √([(-1-4)² + (5-0)²] = √[(-5)² + 5²] = √(25 + 25) = √50

If a² + b² = c², then the triangle is a right triangle.

(√40)² + (√10)² = (√50)²

40 + 10 = 50

50 = 50

It is a right triangle.