Question

The area of a square ABCD is 16cm², find the area of the square formed by joining the mid points P,Q,R and S of the sides, AB,BC,CD,DA respectively.
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Answer 1

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Answer 2

Given :

• ABCD is square with area = 16 cm²
• P is midpoint of AB , AP = PB
• Q is midpoint of BC , BQ= QC
• R is midpoint of CD , CR = RD
• S is midpoint of DA , DS = SA

To find :

Area of figure PQRS

Formula used :

• Area of square = (side)²
• Area of triangle = (1/2) base × height

Analysis :

• Here, first of all we need to find side of square ABCD.
• After that we will find the area of 4 triangle.
• Area of PQRS , will be area of square minus area of 4 triangles

Let :

Side of square ABCD = x

Solution :

➝ Area of ABCD = x²

➝ 16cm² = x²

➝ x = √(16cm²)

➝ x = ± 4cm

{ As side cannot be negative }

➝ x = 4cm

➝ AB = BC = CD = DA = 4cm

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From above ,

we know that,

➝ AP = PB = (1/2)AB

➝ AP = PB = 4/2 = 2cm

➝ BQ= QC = BC/2

➝ BQ= QC = 4/2 = 2cm

➝ CR = RD = CD/2

➝ CR = RD = 4/2 = 2cm

➝ DS = SA = AD/2

➝ DS = SA = 4/2 = 2cm

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Now find area of 4 traingle.

All 4 traingle have there ,

• base = 2cm
• height = 2cm

Therefore there area will be equal,

{ So we will find area of one triangle and multiply it by 4 to get area of 4 triangle }

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Area of ∆RSD = (1/2) × RS × SD

Area of ∆RSD = (1/2)×2×2 cm²

Area of ∆RSD = 2cm²

Area of 4 triangle = 4×(Area of ∆RSD)

Area of 4 triangle = 4×2 = 8cm²

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Area of PQRS = Area of square ABCD - Area of 4 triangle

Area of PQRS = 16cm² - 8cm²

Area of PQRS = 8cm²

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ANSWER :

Area of PQRS = 8cm²