Question

If P(-2,4), Q(4,8), R(10, 5) and S(4, 1) are the vertices of a quadrilateral, show that it is a parallelogram.

Answer 1

To prove: PQRS is a parallelogram.

The quadrilateral is a parallelogram if opposite sides of quadrilateral are equal in length and parallel to each other

Consider PS and QR as opposite sides of quadrilateral

And PQ and RS as opposite sides of quadrilateral

The slope of a line is given by `(y_2 - y_1)/(x_2 - x_1)` where `(x_1,y_1) & (x_2,y_2)` are points on line

Checking for PS and QR

the slope of PS = `(1 - 4)/(4 - (-2)) = (-3)/6 = -1/2`

slope of QR = `(5 - 8)/(10 - 4) = (-3)/6 = -1/2`

Slope of PS = Slope of QR…(i)

Therefore, PS parallel to QR

Length of PS using distance formula

`"PS" = sqrt((4 - (-2))^2 + (1 - 4)^ = sqrt(36 + 9) = sqrt(9 xx 5) = 3sqrt(5)

Length of QR

`"QR" = sqrt((10 - 4)^2 + (5 - 8)^2) = sqrt(36 + 9) = sqrt(9 xx 5) = 3sqrt(5)`

Length of PS = Length of QR…(ii)

Checking for PQ and RS

Slope pf PQ = `(8 - 4)/(4 - (-2)) = 4/6 = 2/3`

Slope of RS = `(1 - 5)/(4 - 10) = (-4)/(-6) = 2/3`

Slope of PQ = Slope of RS…(iii)

Therefore, PQ parallel to RS

Length of PQ using distance formula

`"PQ" = sqrt((4 - (-2))^2 + (8-4)^2) = sqrt(36 + 16) = sqrt(52)`

Length of RS

`"QR" = sqrt((4 - 10)^2 + (1 - 5)^2) = sqrt(36 + 16) = sqrt(52)`

Length of PQ = Length of RS…(iv)

Using equations (i),(ii),(iii) and (iv) we conclude that quadrilateral PQRS is a parallelogram

Hence proved