Question

The height of an equilateral triangle is \[\sqrt 6 cm\] long. Its area is
A) \[3\sqrt 3 c{m^2}\]
B) \[2\sqrt 3 c{m^2}\]
C) \[2\sqrt 2 c{m^2}\]
D) \[6\sqrt 2 c{m^2}\]

Answer 1

Hint: In this question first, we find the length of a side of an equilateral triangle using the formula \[h = \dfrac{{\sqrt 3 }}{2} \times a;\] Where ‘a’ is the side and ‘h’ is the height of an equilateral triangle. After that use the obtained length of the side of an equilateral triangle to find the area of an equilateral triangle using the formula \[\dfrac{{\sqrt 3 }}{4} \times {a^2}\].

Complete step by step solution: We have given height (altitude) of an equilateral triangle is \[\sqrt 6 cm\].

As we know there is a relation between the height (altitude) and side of an equilateral triangle using that we can find the length of a side of an equilateral triangle.
\[h = \dfrac{{\sqrt 3 }}{2} \times a;\] Where ‘a’ is side and ‘h’ is height of an equilateral triangle.
\[ \Rightarrow \sqrt 6 = \dfrac{{\sqrt 3 }}{2} \times a\]
On cross multiplication we get,
\[ \Rightarrow \dfrac{{\sqrt 6 \times 2}}{{\sqrt 3 }} = a\]
\[ \Rightarrow a = \dfrac{{\sqrt 2 \times \sqrt 3 \times 2}}{{\sqrt 3 }}\]
On simplifying we get,
\[ \Rightarrow a = 2\sqrt 2 \]
Which implies the length of ‘a’ sides of an equilateral triangle is \[2\sqrt 2 \]cm.
Now, we know that
Area of equilateral triangle \[ = \dfrac{{\sqrt 3 }}{4} \times {a^2}\]
Now, put the value of ‘a’ in the above formula we get,
Area \[ = \] \[\dfrac{{\sqrt 3 }}{4} \times {\left( {2\sqrt 2 } \right)^2}\]
On squaring we get,
\[ \Rightarrow \] Area \[ = \dfrac{{\sqrt 3 }}{4} \times 4 \times 2\]
On further simplification we get,
\[ \Rightarrow \] Area \[ = 2\sqrt 3 c{m^2}\]
Thus, the area of an equilateral triangle is \[2\sqrt 3 c{m^2}\]

Hence, option B \[2\sqrt 3 c{m^2}\]is the correct answer.

Note: An equilateral triangle is a triangle in which all three sides are equal. It is also known as an equiangular; that is, all three internal angles are also congruent to each other and are each \[{60^ \circ }\]. It is also referred to as a regular triangle. Altitude or height of a triangle is the perpendicular line from the side of the triangle to its opposite vertex. For an equilateral triangle, all the altitudes will be equal are the perpendicular bisectors of the sides. Circumcircle is the circle that connects all the vertices of a triangle. Its centre is known as circumcenter and is determined by the point of intersection of the perpendicular bisectors of the sides of the triangle. Orthocenter of a triangle is defined as the point where all the altitudes of a triangle meet. For an equilateral triangle, the altitudes are also the perpendicular bisectors of the side. So, the circumcenter and orthocenter of an equilateral triangle are the same.