Question

Prashant had a plot of land in the shape of a quadrilateral. He constructed his house in the middle by joining the midpoints of the four sides of the land and used the remaining four portions at the four ends for different purposes, like a small garden, swimming pool, etc.

1. What type of a quadrilateral is PQRS?
2. What are the lengths of adjacent sides of the quadrilateral PQRS, if their ratio is 1 : 2 and the perimeter of the quadrilateral is 180 m?
3. In quadrilateral PQRS, if ∠PSQ = 30° and ∠QRS = 110°, find ∠SQP.

Answer 1

(i)

By the Mid-Point Theorem, the line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. Applying this to ΔABD and ΔCBD with diagonal BD, we find that SP || BD (SP = `1/2` BD) and RQ || BD (RQ = `1/2` BD). Thus, SP || RQ and SP = RQ, which proves that PQRS is a parallelogram.

(ii)

Let the lengths of the adjacent sides be x and 2x. The perimeter of a parallelogram is given by twice the sum of its adjacent sides:

Perimeter = 2(x + 2x) = 180 m
2(3x) = 180
6x = 180
x = 30 m

Therefore, the lengths of the adjacent sides are:

Side 1 = 30 m

Side 2 = 2 × 30
= 60 m

(iii)

In a parallelogram, opposite angles are equal. Since PQRS is a parallelogram, ∠SPQ = ∠QRS = 110°.

Now, considering ΔSPQ, the sum of all interior angles is 180°:

∠PSQ + ∠SPQ + ∠SQP = 180°

30° + 110° + ∠SQP = 180°

140° + ∠SQP = 180°

∠SQP = 180° − 140° = 40°