Question

the area of equilateral triangle is 20root3m^2 find its altitude​

Answer 1

» To Find :

The height of the triangle .

» Given :

• Area of the Equilateral Triangle = 20√3 m².

» We Know :

Area of a Equilateral Triangle :

$\sf{\underline{\boxed{A_{t} = \dfrac{\sqrt{3}a^{2}}{4}}}}$

Height of a Equilateral Triangle :

$\sf{\underline{\boxed{A_{t} = \dfrac{\sqrt{3}a}{2}}}}$

Where , a is the side of the triangle.

» Concept :

To Find the height of the Equilateral triangle , first we have to find the side of the triangle.

To find the side of the triangle , we can apply the formula for area of the Equilateral Triangle .

Side of the triangle :

• Area = 20√3 m²

Formula for Area of a Equilateral Triangle :

$\sf{\underline{\boxed{A_{t} = \dfrac{\sqrt{3}a^{2}}{4}}}}$

Putting the value in it ,we get :

$\sf{\Rightarrow 20\sqrt{3} = \dfrac{\sqrt{3}a^{2}}{4}}$

$\sf{\Rightarrow 20\sqrt{3} \times 4 = \sqrt{3}a^{2}}$

$\sf{\Rightarrow 80\sqrt{3} = \sqrt{3}a^{2}}$

$\sf{\Rightarrow \dfrac{80\sqrt{3}}{\sqrt{3}} = a^{2}}$

$\sf{\Rightarrow \dfrac{80\cancel{\sqrt{3}}}{\sqrt{\cancel{3}}} = a^{2}}$

$\sf{\Rightarrow 80 = a^{2}}$

$\sf{\Rightarrow \sqrt{80} = a}$

$\sf{\Rightarrow 4\sqrt{5} m = a}$

Hence , the side of the Equilateral triangle is 4√5 m.

Now ,with this information we can find the Height of the triangle.

» Solution :

Given :

• Side of the triangle = 4√5 m

Formula :

$\sf{\underline{\boxed{A_{t} = \dfrac{\sqrt{3}a}{2}}}}$

By substituting the values in it ,we get :

$\sf{\Rightarrow A_{t} = \dfrac{\sqrt{3} \times 4\sqrt{5}}{2}}$

$\sf{\Rightarrow A_{t} = \dfrac{\sqrt{3} \times \cancel{4}\sqrt{5}}{\cancel{2}}}$

$\sf{\Rightarrow A_{t} = \sqrt{3} \times 2\sqrt{5}}$

$\sf{\Rightarrow A_{t} = 2\sqrt{15} m}$

Hence, the height of the triangle is 4√5 m.

» Additional information :

• Area of a parallelogram = base × height

• Diagonal of a Cube = √3a

• Surface area of a Cube = 6(a)²

• curved surface area of a Cuboid = 4(a)²

Answer 2

Area of equilateral triangle = 20√3 m²

$\rm Area \: of \: equilateral \: triangle = \boxed{ \frac{ \sqrt{3} }{4} \times side² }$

$\frac{ \sqrt{3} }{2} \times side = 20\sqrt{3}$

$side = \frac{2}{ \sqrt{3}} \times 20 \sqrt{3} \\ side = 20 \times 2 \\ side = 40 \: m$

$\rm Another \: formala \: fo r \: area \: of \: triangle = \boxed{ \frac{1}{2} \times base \times altitude}$

$\red{base = side }$

$20 \sqrt{3} = \frac{1}{2} \times 40 \times altitude$

$20 \sqrt{3} = 20 \times altitude \\ \frac{20 \sqrt{3} }{20} = altitude \\ \sqrt{3} \: m = altitude$