Question

Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of a parallelogram.

Answer 1

The four given points are ( −2,−1 ), ( 4,0 ), ( 3,3 ) and ( −3,2 ).

Let P, Q, R, and S be the four given points are ( −2,−1 ), ( 4,0 ), ( 3,3 ) and ( −3,2 ) respectively.

The formula of slope of a line passing through two different points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Let m PQ , m QR , m RS , m PS be the slope of the line segment PQ, QR, RS, PS respectively.

Substitute the values ( −2,−1 ), and ( 4,0 ) for ( x 1 , y 1 ) and ( x 2 , y 2 ) in equation (1) to obtain the slope of line segment PQ.

m PQ = 0+1 4+2 = 1 6

Substitute the values ( 3,3 ), and ( −3,2 ) for ( x 1 , y 1 ) and ( x 2 , y 2 ) in equation (1) to obtain the slope of line segment RS.

m RS = 2−3 −3−3 = −1 −6 = 1 6

The condition for two lines if they are parallel,

slope of first line=slope of second line

It is observed that slope of line segment PQ and line segment QR are equal to each other.

m PQ = m RS

So, PQ and RS are parallel to each other.

Similarly, substitute the values ( 4,0 ), and ( 3,3 ) for ( x 1 , y 1 ) and ( x 2 , y 2 ) in equation (1) to obtain the slope of line segment QR.

m QR = 3−0 3−4 = 3 −1 =−3

Substitute the values ( −2,−1 ), and ( −3,2 ) for ( x 1 , y 1 ) and ( x 2 , y 2 ) in equation (1) to obtain the slope of line segment QR.

m PS = 2+1 −3+2 = 3 −1 =−3

It is observed that slope of line segment QR and line segment PS are equal to each other.

m QR = m PS

So, QR and PS are parallel to each other.

The opposite pair of sides of a parallelogram are parallel to each other.

So, the both pair of opposite sides is parallel to each other.

Hence, the vertices ( −2,−1 ), ( 4,0 ), ( 3,3 ) and ( −3,2 ) are the vertices of a parallelogram.