In an equilateral triangle of side 24 cm, find the length of the altitude.

To find the length of the altitude in an equilateral triangle, we can use the Pythagorean theorem.

Step 1: Recall that in an equilateral triangle, all sides are equal. In this case, the side length is given as 24 cm.

Step 2: Draw an altitude from one vertex of the triangle to the opposite side. This altitude will bisect the base and form a right triangle.

Step 3: Let's label the length of the altitude as 'h' and the length of half the base as 'b'. Since the triangle is equilateral, the altitude bisects the base into two equal parts, so 'b' is equal to half of the side length, which is 24/2 = 12 cm.

Step 4: Now, we can apply the Pythagorean theorem to find the length of the altitude. The theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Step 5: In our case, the altitude 'h' is the hypotenuse, and the two sides are 'b' and the unknown side, which is also 'h'. So, we have the equation h^2 = b^2 + h^2.

Step 6: Simplifying the equation, we get h^2 = 12^2 + h^2.

Step 7: Solving for h, we subtract h^2 from both sides of the equation: 0 = 12^2.

Step 8: Taking the square root of both sides, we find that h = 12 cm.

Therefore, the length of the altitude in the equilateral triangle is 12 cm.