Question

Diagonals AC and BD of square ABCD intersect at P. Coordinates of points B and D are (9, −2) and (1, 6) respectively.

1. Find the co-ordinates of point P.
2. Find the length of the side of the square.

Answer 1

Given: B = (9, −2)

D = (1, 6)

(i) In a square, the diagonals bisect each other. Therefore, the intersection point P is the midpoint of the diagonal BD.

Using the midpoint formula for B (9, −2) and D (1, 6)

P = `((x_1 + x_2)/2, (y_1 + y_2)/2)`

= `((9 + 1)/2, (-2 + 6)/2)`

= `(10/2, 4/2)`

= (5, 2)

(ii) The length of the diagonal BD can be found using the distance formula:

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

= `sqrt((1 - 9)^2 + (6 - (-2))^2)`

= `sqrt((-8)^2 + (6 + 2)^2)`

= `sqrt((-8)^2 + (8)^2)`

= `sqrt(64 + 64)`

= `sqrt(128)`

= `8sqrt(2)`

The relation between the side and the diagonal in a square:

Diagonal = `"side" xx sqrt2`

`(8sqrt(2)) = "side" xx sqrt2`

side = `(8sqrt(2))/(sqrt2)`

side = 8