Mathematics

Calculate the area of the shaded region.

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From figure,

Area of △AOB = $\dfrac{1}{2}$ × base × height

= $\dfrac{1}{2}$ × AO × OB

= $\dfrac{1}{2}$ × 12 × 5

= 30 cm².

In right angle triangle AOB,

Using pythagoras theorem,

⇒ AB² = AO² + OB²

⇒ AB² = 12² + 5²

⇒ AB² = 144 + 25

⇒ AB² = 169

⇒ AB = $\sqrt{169}$ = 13 cm.

In △ABC,

Let a = BC = 14 cm, b = AC = 15 cm and c = AB = 13 cm.

s = $\dfrac{a + b + c}{2} = \dfrac{14 + 15 + 13}{2} = \dfrac{42}{2}$ = 21 cm.

By Heron's formula,

Area = $\sqrt{s(s - a)(s - b)(s- c)}$

Substituting values we get,

$= \sqrt{21(21 - 14)(21 - 15)(21 - 13)} \\[1em] = \sqrt{21 \times 7 \times 6 \times 8} \\[1em] = \sqrt{7056} \\[1em] = 84 \text{ cm}^2.$

Area of shaded region = Area of △ABC - Area of △AOB

= 84 - 30 = 54 cm².

Hence, area of shaded region = 54 cm².

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