Question

Find the altitude of an equilateral triangle of sides equal to 10 cm.

Answer 1

Hint: We had to draw an altitude from one vertex of the equilateral triangle to the opposite side of the equilateral triangle and then we will apply the Pythagorean theorem on one of the two triangles formed by the altitude of the equilateral triangle.

Complete Step-by-Step solution:
As we know that the equilateral triangle is a triangle whose lengths of all sides are equal.
So, let ABC be an equilateral triangle whose all the sides are equal to 10cm.
So, now as we know that the altitude or height of the triangle is a line segment formed from any vertex that is perpendicular to the opposite side.
But if the triangle is equilateral then altitude also bisects the side on which it is perpendicular.
So, let us draw a triangle ABC in which AD is altitude.

As the triangle ABC equilateral triangle. So, CD = DB. And AD will be the altitude.
Now as we can see from the above figure that triangle ADB is the right-angle triangle right-angled at D. So, we can apply Pythagorean theorem to find the value of AD.
According to the Pythagorean theorem \[{\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{Base}}} \right)^2}\].
So, applying Pythagorean theorem in triangle ADB. We get,
\[{\left( {AB} \right)^2} = {\left( {AD} \right)^2} + {\left( {{\text{DB}}} \right)^2}\]
\[{\left( {10} \right)^2} = {\left( {AD} \right)^2} + {\left( 5 \right)^2}\]
\[100 = {\left( {AD} \right)^2} + 25\]
Subtract 25 from both the sides of the above equation and then take the square root. We get,
\[AD = \sqrt {75} = 5\sqrt 3 \]cm
Hence, the length of the altitude of the triangle whose side is equal to 10 cm and the triangle is equilateral triangle will be \[5\sqrt 3 \]cm.

Note: Whenever we come up with this type of problem then we should remember that the altitude of the triangle is the line segment joining the one vertex of the triangle and is perpendicular to the side opposite to it. But if the triangle is equilateral then it also bisects the side opposite to the vertex. So, here the triangle is an equilateral triangle. So, we will get that DB equal to 5 cm and then we will apply Pythagorean theorem in triangle ADB to find the value of the altitude of the triangle AD.