Question

prove that the points (2 -2),(-2 1) and (5 2) are the vertices of a right angled triangle. also find the area of triangle

Answer 1

As we know in right angled triangled

Two lines make an angle 90

So lets find slope between two points

(2,-2) ( -2,1)

slope( m1) = ( 1 -(-2))/ (-2-2) = 3/-4 =-3/4

Slope (m2) between (-2,1) and (5,2)

m2 = ( 2 -1)/ ( 5-(-2)) =1/7

Slope (m3) between (2,-2) and (5,2)

m3 = ( 2 -(-2))/ (5-2) = 4/3

Lets find m1 × m3

m1= -3/4

m3= 4/3

m1 × m3 = -3/4 × 4/3 = -1

As For pendicular lines product of slope is -1

Hence its right angled triangle

Lets find distance between (2,-2) and (-2,1)

√ ( 2 -(-2))^2 + ( -2-1)^2 = √4^2 +3^2 = √16+9 = √25 = 5

Now distance between (2,-2) and (5,2)

√( 2 -5)^2 + ( -2 -2)^2 = √3^2 +4^2 = √25 = 5

Area = 1/2 × base × height

Base = height = 5

So area = 1/2 × 5 × 5 = 25/2 = 12.5