Question

a equilateral triangle of side 10 cm is provided... find its area....​

Answer 1

Answer:

Step-by-step explanation:

Is it its height or area

Anyway

Its area is given by formula

Root3a÷4

Area=_/3×5÷2

Height is given by _/3a÷2

Heigjt =5_/3

Note _/ means root

Answer 2

We know about the Heron's formula, i.e, $\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$

We also know the formula for the area of equilateral triangle, that is derived from the Heron's Formula, i.e., $\text{A} = \dfrac{\sqrt{3}}{4}\text{s}^\text{2}$

So as we know the formula for find the area of a n Equilateral Triangle, let's start solving the question;

So, here we are given the value of s (Side of the triangle), i.e., 10 cm

As we have the formula as well as the size of the side, let's plug in the value into the formula;

$\text{A} = \dfrac{\sqrt{3}}{4}\times\text{10}^\text{2}$

Let's start with the squaring of 10;

$\text{A} = \dfrac{\sqrt{3}}{4}\times\text{100}$

Simplify. :)

$\text{A} = \dfrac{\sqrt{3}}{\cancel{4}}\times\cancel{\text{100}}\\\\\implies \text{A} = \dfrac{\sqrt{3}}{1}\times\text{25}\\\\\implies \text{A} = \sqrt{\text{3}}\times\text{25} = \bold{25\sqrt{3}}$

Therefore, the area of the equilateral triangle is $25\sqrt{3}$.

Now, the question has one more part saying "Find the height of the triangle"

So to start this question we'll need to derive the formula for the height;

So first let's divide the equilateral triangle into two equal halves, i.e., drawing its altitude. (see the image attached)

Now lets suppose only the half of the triangle.

So, Now it looks like a right angled triangle;

Thereby applying the Pythagoras Theorem:

$\implies (a)^2=(h)^2+\Big(\dfrac{a}{2}\Big)^2\\\\\implies (h)^2=\bigg(\dfrac{3}{4}\bigg)a^2\\\\\implies \bold{h=\dfrac{\sqrt{3}}{2}a}$

Now we've derived the formula for the height, So, let's plug in the value;

$h=\dfrac{\sqrt{3}}{2}\times10$

Simplify. :)

$\sqrt{3}\times5\\\implies \bold{5\sqrt{3}}$

Therefore, the height of the triangle is $5\sqrt{3}$.