Question

Prove that (2, -2), (-2, 1) and (5, 2) are the vertices of a right angled triangle. Find the area of the triangle and the length of the hypotenuse.

Answer 1

Let A(2, -2), B(-2, 1) and C(5, 2) be the given points

⇒ AB=√(16-9)

⇒ AB=√(25)

Since, AB²+AC²=BC²

Therefore, ABC is a right angled triangle.

Length of the hypotenuse BC=√(50)=5√2

Area of ΔABC=(1/2)xABxAC

=(1/2)x√25x√25

=25/2 sq. units.

Answer 2

Let A (2,-2) B (-2,1) and c (5,2)

now ,
use distance formula ,
AB = {(2+2)^2 + (-2-1)^2}^1/2= 5
BC ={(5+2)^2 +(2-1)^2}^1/2 = (50)^1/2
CA = {(5-2)^2 +(2 + 2)^2 }^1/2 = 5
now ,
AB^2 =25
BC^2 =50
CA^2 = 25
you can see ,
AB^2 +CA^2 =BC^2
according to Pythagoras theorem ,
ABC is right angle triangle .
Hence proved