Question

The points A(–1, –2), B(4, 3), C(2, 5) and D(–3, 0) in that order form a rectangle.

पर्याय

• True

• False

Answer 1

This statement is True.

Explanation:

Distance between A(–1, –2), B(4, 3),   

AB = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

AB = `sqrt((4 + 1)^2 + (3 + 2)^2` 

= `sqrt(5^2 + 5^2)`

= `sqrt(25 + 25)`

= `5sqrt(2)`

Distance between C(2, 5) and D(–3, 0),

CD = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

CD = `sqrt((-3 - 2)^2 + (0 - 5)^2`

= `sqrt((-5)^2 + (-5)^2)`

= `sqrt(25 + 25)`

= `5sqrt(2)`

Distance between A(–1, –2) and D(–3, 0),

AD = `sqrt((-3 + 1)^2 + (0 + 2)^2`

= `sqrt((-2)^2 + 2^2)`

= `sqrt(4 + 4)`

= `2sqrt(2)`

And distance between B(4, 3) and C(2, 5),

BC = `sqrt((4 - 2)^2 + (3 - 5)^2`

= `sqrt(2^2 + (-2)^2)`

= `sqrt(4 + 4)`

= `2sqrt(2)`

We know that, in a rectangle, opposite sides and equal diagonals are equal and bisect each other.

Since, AB = CD and AD = BC

Also, distance between A(–1, –2) and C(2, 5),

AC = `sqrt((2 + 1)^2 + (5 + 2)^2`

= `sqrt(3^2 + 7^2)`

= `sqrt(9 + 49)`

= `sqrt(58)`

And distance between D(–3, 0) and B(4, 3),

DB = `sqrt((4 + 3)^2 + (3 - 0)^2`

= `sqrt(7^2 + 3^2)`

= `sqrt(49 + 9)`

= `sqrt(58)`

Since, diagonals AC and BD are equal.

Hence, the points A(–1, – 2), B(4, 3), C(2, 5) and D(–3, 0) form a rectangle.