From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. The lengths of the perpendiculars are 14 cm, 10 cm, and 6 cm.
The area of the triangle is $300 \sqrt{m} c m^{2}$ where $m$ is _______

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Solution

$L e t P b e a n y p o i n t o n t h e i n t e r i o r o f a n e q u i l a t e r a l t r i a n g l e A B C . J o i n P t o a l l 3 v e r t i c e s o f t h e t r i a n g l e . O n d o i n g s o, t h e t r i a n g l e g e t s d i v i d e d i n t o t h r e e s m a l l e r t r i a n g l e s w i t h h e i g h t s 6 c m, 10 c m & 14 c m r e s p e c t i v e l y . L e t t h e s i d e o f t h e t r i a n g l e b e y . T h e a r e a o f a △ = \frac{1}{2} \times b a s e \times h e i g h t A r e a o f △ P A B = \frac{1}{2} \times y \times 14 = 7 y A r e a o f △ P B C = \frac{1}{2} \times y \times 10 = 5 y A r e a o f △ P C A = \frac{1}{2} \times y \times 6 = 3 y S o, t h e t o t a l a r e a o f t h e △ A B C = △ P A B + △ P B C + △ P C A = 7 y + 5 y + 3 y = 15 y - - - - - - - - - - (1) N o w w e k n o w, t h e a r e a o f a n e q u i l a t e r a l t r i a n g l e = \frac{\sqrt{3}}{4} \times y^{2} (w h e r e y i s t h e s i d e o f t h e t r i a n g l e) . - - - - (2) H e n c e e q u a t i n g t h e a r e a s 1 & 2 w e g e t, 15 y = \frac{\sqrt{3}}{4} \times y^{2} \Rightarrow 15 = \frac{\sqrt{3}}{4} \times y [c a n c e l l i n g y f r o m b o t h s i d e s] \Rightarrow 15 \times 4 = \sqrt{3} \times y \Rightarrow y = \frac{60}{\sqrt{3}} ∴ T h e a r e a o f t h e t r i a n g l e = 15 y = 15 \times \frac{60}{\sqrt{3}} = 15 \times \frac{20 \times 3}{\sqrt{3}} = 300 \sqrt{3}$

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From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. The lengths of the perpendiculars are 14 cm, 10 cm, and 6 cm. The area of the triangle is 300√mcm2 where m is _______

From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. The lengths of the perpendiculars are 14 cm, 10 cm, and 6 cm.
The area of the triangle is $300 \sqrt{m} c m^{2}$ where $m$ is _______