Question

What is the area (in
c
m
2
) of the quadrilateral formed by joining the midpoints of the sides of a square of area 144
c
m
2
?​

Answer 1

Given:

What is the area (in cm2) of the quadrilateral formed by joining the midpoints of the sides of a square of area 144 cm²?

To find:

The area (in cm2) of the quadrilateral

Solution:

Finding the length of the side of the given square:

Let ABCD be the square with each side = "a" cm.

We know,

$\boxed{\bold{Area\:of\:a\:square = side^2}}$

The area of the square ABCD = 144 cm²

∴ a² = 144

⇒ a = √144

⇒ a = 12 cm

Finding the area of the quadrilateral:

Let PQRS be the quadrilateral formed by joining the midpoints of the square ABCD and a midpoint will divide each side of ABCD into half.

∴ The length of each side of PQRS = $\frac{1}{2} \times a = \frac{1}{2} \times 12 = 6\: cm$

Also, let us consider rt Δ APS, by using the Pythagoras theorem, we get

$PS^2 = AP^2 + AS^2$

$\implies PS = \sqrt{AP^2 + AS^2 }$

$\implies PS = \sqrt{6^2 + 6^2 }$

$\implies PS = \sqrt{72 }$

$\implies PS = 8.48\:cm$

Similarly, PQ = QR = RS = 8.48 cm

∴ PQRS is also a square with each side = 8.48 cm

Now,

The area of the square PQRS is,

= side²

= 8.48²

= 71.91 cm²

rounding off to its nearest whole number

≈ 72 cm²

Thus, the area of the quadrilateral formed by joining the midpoints of the sides of a square of area 144 cm² is → 72 cm².

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Answer 2

Answer:

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Step-by-step explanation:

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