Question

In the rectangle ABCD, AB = 12 cm, BC = 7 cm. P and R are mid-points of AB and CD. AS = BQ = 2.5 cm. Find the perimeter of PQRS.

Answer 1

Given:

Rectangle ABCD,

AB = 12 cm

BC = 7 cm

P and R are midpoints of AB and CD.

AS = BQ = 2.5 cm

To find: The perimeter of quadrilateral PQRS.

Calculation:

1. Use coordinates:

Place the rectangle on the coordinate plane:

A = (0, 0)

B = (12, 0)

C = (12, 7)

D = (0, 7)

Now place the relevant points:

P is midpoint of AB:

P = `((0 +12)/2,0)` = (6, 0)

R is midpoint of CD:

R = `((0 + 12)/2,7)` = (6, 7)

S lies on AD such that AS = 2.5 cm.

Since AD is vertical from (0, 0) to (0, 7), S = (0, 2.5)

Q lies on BC such that BQ = 2.5 cm

Since BC is vertical from (12, 0) to (12, 7),

Q = (12, 2.5)

2. Find lengths of sides of PQRS:

Use the distance formula:

PQ = `sqrt((12 - 6)^2 + (2.5 - 0)^2`

= `sqrt(36 + 6.25)`

= `sqrt(42.25)`

= 6.5 cm

QR = `sqrt((12 - 6)^2 + (7 - 2.5)^2`

= `sqrt(36 + 20.25)`

= `sqrt(56.25)`

= 7.5 cm

RS = `sqrt((6 - 0)^2 + (7 - 2.5)^2`

= `sqrt(36 + 20.25)`

= `sqrt(56.25)`

= 7.5 cm

SP = `sqrt((6 - 0)^2 + (0 - 2.5)^2`

= `sqrt(36 + 6.25)`

= `sqrt(42.25)`

= 6.5 cm

3. Total perimeter of PQRS:

PQ + QR + RS + SP

= 6.5 + 7.5 + 7.5 + 6.5

= 28 cm