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प्रश्न
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.
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उत्तर
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
