Question

find the area of shaded region in which AC=33cm,BC=24cm,AD=12cm and BD =5cm and angle ADB=90°​

Answer 1

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Answer 2

Answer:

I have attached the given diagram.

We are given the following sides of the triangle,

AC = 33 cm

BC = 24 cm

AD = 12 cm

BD = 5 cm

∠ADB = 90°

Also, we have to take √35 = 5.91

Now, We have to find the area of the yellow coloured shape. Clearly it can be found by subtracting the area of ∆ADB from the area of ∆ABC.

But we need the third side of ∆ABC, i.e., AB which can be found using the Pythagoras theorem in ∆ADB,

In ∆ADB,

using Pythagoras theorem,

⇒ Hypotenuse² = Base² + Perpendicular²

⇒ AB² = BD² + AD²

⇒ AB² = 5² + 12²

⇒ AB² = 25 + 144

⇒ AB² = 169

⇒ AB = 13

Now, In ∆ABC using heron's formula, we have

a = AB = 13 cm

b = BC = 24 cm

c = AC = 33 cm

Also,

Semi-perimeter, s = (a + b + c)/2 = 35 cm

Now,

⇒ ar(∆ABC) = √{ s(s - a)(s - b)(s - c) }

⇒ ar(∆ABC) = √{ 35(35 - 13)(35 - 24)(35 - 33) }

⇒ ar(∆ABC) = √( 35 × 22 × 11 × 2 )

⇒ ar(∆ABC) = 22√35

⇒ ar(∆ABC) = 130.02 cm²

Similarly, In ∆ADB, Since it is a right angled triangle , so

⇒ ar(∆ADB) = 1/2 × base × height

⇒ ar(∆ADB) = 1/2 × BD × AD

⇒ ar(∆ADB) = 1/2 × 5 × 12

⇒ ar(∆ADB) = 5 × 6

⇒ ar(∆ADB) = 30 cm²

So, Area of shaded region (yellow coloured region)

⇒ ar(∆ABC) - ar(∆ADB)

⇒ 130.02 - 30

⇒ 100.02 cm²