Question

Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.

Answer 1

The three given points are ( 4,4 ),( 3,5 ) and ( −1,−1 ).

The formula for slope m of a non-vertical line passing through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Let, the slope of the line passing through the points ( 4,4 ) and ( 3,5 ) is m 1 .

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 4,4 ) and ( 3,5 ) respectively in equation (1).

m 1 = 5−4 3−4 =−1

Let, the slope of line passing through the points ( 3,5 ) and ( −1,−1 ) is m 2 .

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 3,5 ) and ( −1,−1 ) respectively in equation (1).

m 2 = −1−5 −1−3 = −6 −4 = 3 2

Similarly let, the slope of line passing through the points ( −1,−1 ) and ( 4,4 ) is m 3 .

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( −1,−1 ) and ( 4,4 ) respectively in equation (1)

m 3 = 4+1 4+1 = 5 5 =1

The condition for perpendicularity of two is given by,

( slope of first line )×( slope of second line )=−1

It is observed that the product of slope m 1 and m 3 is equal to −1.

m 1 × m 3 =1×−1 =−1

This shows that the line segment joining ( 4,4 ) and ( 3,5 ) with the line segment ( −1,−1 ) and ( 4,4 ) are perpendicular to each other.

Also as two line segment are perpendicular, the right angle formed at common point ( 4,4 ).

Thus, the vertices ( 4,4 ), ( 3,5 ) and ( −1,−1 ) formed a right angle triangle with right angle at ( 4,4 ).