Question

A point P is selected inside an equilateral triangle ABC. Sum of the perpendiculars PD, PE, PF from P on AB, BC, CA respectively is 2020. What is the value of [tex] \frac{Altitude\: of\:ΔABC} {2020}.

No random answer. Answer only if you know, along with proper calculation. ​

Answer 1

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✤ Required Answer:

✒ GiveN:

• P is any point inside the triangle.
• PD, PE and PF are the perpendiculars from point P to sides BC, AC and AB
• Sum of the perpendiculars = 2020

✒ To FinD:

• Value of altitude of △ABC ?

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✤ How to solve?

How to do the question is what we will see the solution directly, Just one thing is the area of triangle.

• When height/altitude and corresponding base is given, Then formula for area of △:

$\large{ \boxed{ \sf{ar ( \triangle) = \frac{1}{2} \times Base \times Height(Altitude)}}}$

☄ So, Let's solve this question..m

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✤ Solution:

⬆️ Refer to the attachment...

We have,

• PD, PE and PE as perpendiculars
• Corresponding base BC, CA and AB

Then,

➝ ar(△PAB) = 1/2 × PF × AB

➝ ar(△PBC) = 1/2 × PD × BC

➝ ar(△PCA) = 1/2 × PE × CA

As, the sides are same, let it be x

➝ ar(△ABC) = 1/2 × Altitude(To find) × x

Now, We know that,

➝ ar(△ABC) = ar(△PAB) + ar(△PBC) + ar(△PCA)

➝ 1/2 × altitude × x = 1/2 × PF × x + 1/2 × PD × x + 1/2 × PE × x

Taking common 1/2 × x from both sides,

➝ 1/2 × x × (Altitude) = 1/2 × x(PF + PD + PE)

➝ Altitude of △ABC = PF + PD + PE

Given,

• Sum of PD, PE and PF = 2200

Therefore,

➝ Altitude of △ABC = 2200

☃️ Hence, solved !!

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