Question

ABCD is a rectangle formed by joining the points A(–1, –1), В(–1, 4), С(5, 4) and D(5, –1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Show that the diagonals of the quadrilateral PQRS bisect each other.

Answer 1

Given: A(–1, –1), B(–1, 4), C(5, 4), D(5, –1). P, Q, R, S are mid-points of AB, BC, CD, DA respectively.

Step-wise calculation:

1. Find mid-points:

P = midpoint of AB = `((-1 + (-1))/2, (-1 + 4)/2) = (-1, 3/2)`.

Q = midpoint of BC = `((-1 + 5)/2, (4 + 4)/2) = (2, 4)`.

R = midpoint of CD = `((5 + 5)/2, (4 + (-1))/2) = (5, 3/2)`.

S = midpoint of DA = `((5 + (-1))/2, (-1 + (-1))/2) = (2, -1)`.

2. Coordinates of diagonals of PQRS:

Diagonal PR joins `P(-1, 3/2)` and `R(5, 3/2)`.

Midpoint of PR = `((-1 + 5)/2, (3/2 + 3/2)/2) = (2, 3/2)`.

Diagonal QS joins Q(2, 4) and S(2, –1).

Midpoint of QS = `((2 + 2)/2, (4 + (-1))/2) = (2, 3/2)`.

3. Compare midpoints: Midpoint (PR) = `(2, 3/2)` = Midpoint (QS).

Since the midpoints of the two diagonals PR and QS coincide, the diagonals of quadrilateral PQRS bisect each other.