Let ∆ABC an equilateral triangle of side a cm each
AD is the altitude on side BC
to find:AD=?
In right ∆ADB by Pythagoras th.
AB=a,
BD=a/2
AB²=AD²+BD²
AD²=AB²-BD²=a²-(a/2) ²=a²-a²/4=3a²/4
AD=√3a²/4=a√3/2cm

Length of an Altitude in an Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length, and all three angles are equal to 60 degrees. In this question, we are asked to find the length of an altitude in an equilateral triangle with a side length of 'a' cm.

Definition of an altitude

An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side (base) or its extension. In an equilateral triangle, all three sides are equal in length, and the altitude divides the triangle into two congruent right-angled triangles.

Properties of an equilateral triangle
- All three sides are equal in length.
- All three angles are equal to 60 degrees.
- The altitude divides the triangle into two congruent right-angled triangles.
- The altitude bisects the base, dividing it into two equal segments.

Finding the length of the altitude

To find the length of the altitude in an equilateral triangle, we can use the Pythagorean theorem. Let's consider the equilateral triangle ABC with side length 'a' cm.

1. Draw an altitude from vertex A to the opposite side BC, and let D be the point of intersection.

2. The altitude AD divides the base BC into two equal segments, BD and CD.

3. Since the triangle ABC is equilateral, all three angles are equal to 60 degrees.

4. Triangle ABD is a right-angled triangle, with angle B = 90 degrees.

5. Using the Pythagorean theorem, we can find the length of the altitude AD.

- The hypotenuse AB = a cm (side length of the equilateral triangle).
- The base BD = (1/2) * a cm (half of the side length).
- The altitude AD = ?

Applying the Pythagorean theorem:
AD^2 + BD^2 = AB^2

Substituting the known values:
AD^2 + (1/2)^2 * a^2 = a^2

Simplifying the equation:
AD^2 + (1/4) * a^2 = a^2

Rearranging the equation:
AD^2 = a^2 - (1/4) * a^2
AD^2 = (3/4) * a^2

Taking the square root of both sides:
AD = sqrt((3/4) * a^2)
AD = (sqrt(3)/2) * a

Thus, the length of the altitude in an equilateral triangle with side length 'a' cm is given by the formula:
AD = (sqrt(3)/2) * a

Conclusion

In an equilateral triangle, the length of the altitude can be found using the formula AD = (sqrt(3)/2) * a, where 'a' represents the side length of the triangle.