Name: Finding area of triangle using base and height
Uploaded: 2019-07-19
Duration: 10 min 43 s
Description: Do you know that you can find the area of a triangle using its base and height? Well, that is what we'll learn about in this informative video. Learn more.

Question

Prove that sum of the perpendiculars drawn from any interior point of an equilateral triangle to its sides equal to its height.

Accepted Answer

Given: $△ ABC$ is an equilateral.
$AD, BE and CF$ are altitudes of the given triangle.
$OD, OE, OF$ are the perpendiculars drawn to the sides $BC, AC, A B$ from an internal point $O$ .

From the figure:
$ar (△ ABC) = ar (△ OCB) + ar (△ OAB) + ar (△ OAC)$

We know that, area of the triangle is $\frac{1}{2} \times base \times height$ .

So,
$\Rightarrow \frac{1}{2} \times BC \times AD = \frac{1}{2} \times BC \times OD + \frac{1}{2} \times AB \times OF + \frac{1}{2} \times AC \times OE$

Given, $△ ABC$ is equilateral; $AB = BC = AC$ .

$\Rightarrow \frac{1}{2} \times BC \times AD = \frac{1}{2} \times BC \times OD + \frac{1}{2} \times BC \times OF + \frac{1}{2} \times BC \times OE$

$\Rightarrow \frac{1}{2} \times BC \times AD = \frac{1}{2} \times BC \times (OD + OF + OE)$

$\Rightarrow AD = (OD + OE + OF)$

Here, $A D$ is the height of the triangle $ABC$ .

Therefore, height of the triangle is equals to the sum of perpendiculars drawn from an internal point $O$ .

Hence, proved.

Prove that sum of the perpendiculars drawn from any interior point of an equilateral triangle to its sides equal to its height.

Important Questions on Theorems on Areas

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