The two equal sides of an isosceles triangle are each 24 centimeters. If each of the two equal angles measures 60, find the length of the base and the altitude. (leave your answer in radical form).
LOL, but if two angles are 60 then so is the third and it is an equilateral triangle :)
base = 24
12^2 + h^2 = 24^2
h^2 = -(3*4)^2 + (6 *4)^2
h^2 = 4^2 ( -9+36) = 4^2 (27) = 4^2*3^3
h = 4*3 * sqrt 3
h = 12 sqrt 3
To find the length of the base and the altitude of an isosceles triangle, we can use the properties of a 30-60-90 triangle.
In this case, we are given that the two equal sides of the isosceles triangle are each 24 centimeters, and each of the two equal angles measures 60 degrees.
First, let's find the length of the base (the side opposite the 60-degree angle). In an isosceles triangle, the base is always equal to one of the equal sides. So, the length of the base is 24 centimeters.
Next, let's find the length of the altitude (the perpendicular distance from the base to the opposite vertex). In a 30-60-90 triangle, the ratio of the sides is 1:√3:2, with the hypotenuse being the longest side. Since the base of the isosceles triangle (which is 24 centimeters) is opposite the 60-degree angle, it is equivalent to the longer leg of a 30-60-90 triangle.
So, the altitude is given by the formula: altitude = (√3/2) * base.
Plugging in the value of the base (24 centimeters), the formula becomes:
altitude = (√3/2) * 24 centimeters
Simplifying, we have:
altitude = 12√3 centimeters
Therefore, the length of the base is 24 centimeters, and the length of the altitude is 12√3 centimeters.