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A point O is taken inside an equilateral triangle ABC. If $OL ⊥ BC, OM ⊥ AC$ and $ON ⊥ AB$ such that OL = 14 cm, OM = 10 cm and ON = 6 cm, the area of triangle ABC is:

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519.61

c m 2

511.24

c m 2

564.43

c m 2

523.41

c m 2

answer is A.

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Detailed Solution

Given OL = 14 cm, OM = 10 cm and ON = 6 cm.
Question Image

Let the side of triangle = k.
Area of

Δ A B C

is,
Area of

Δ A B C

= Area of

Δ A O B

+ Area of

Δ A O C

+ Area of

Δ B O C

A r e a o f Δ A B C = 12 × k × O N + 12 × k × O M + 12 × k × O L ⇒ A r e a o f Δ A B C = 12 × k O N + O M + O L ⇒ A r e a o f Δ A B C = 12 × k 6 + 10 + 14 ⇒ A r e a o f Δ A B C = 12 × k × 30 ⇒ A r e a o f Δ A B C = 15 k c m 2

Now, area of equilateral triangle is,

15 k = 34 k 2 ⇒ 15 × 43 = k ⇒ k = 603 ⇒ k = 203 c m

Area of triangle is,

A = 15 × 203 c m 2 ⇒ A = 3003 c m 2

⇒

A= 519.61

c m 2

The area of triangle is 519.61

c m 2

.
Hence, option 1 is correct.

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