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).

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To find the length of the base and the altitude of the isosceles triangle, we can use the properties of trigonometry.

First, let's consider the length of the base. Since we have an isosceles triangle with two equal sides measuring 24 centimeters, the base will be the third side.

To find the length of the base, we can use the Law of Cosines, which states that in a triangle with sides a, b, and c and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab*cos(C)

In our case, since the triangle is isosceles, both angles opposite the equal sides are 60 degrees. We can substitute the known values into the equation:

b^2 = 24^2 + 24^2 - 2 * 24 * 24 * cos(60)
b^2 = 576 + 576 - 576
b^2 = 576

Taking the square root of both sides, we get:
b = √576
b = 24

Therefore, the length of the base of the isosceles triangle is 24 centimeters.

Now, let's find the altitude of the triangle. The altitude is the perpendicular distance from the base to the vertex opposite the base. We can use the property of trigonometry that in a right-angled triangle, the side opposite an angle is equal to the length of the hypotenuse times the sine of that angle.

In our case, we have a right-angled triangle with one side measuring 24 centimeters (the base) and an angle of 60 degrees. Let's call the length of the altitude h.

h = 24 * sin(60)
h = 24 * (√3/2)
h = 12√3

Therefore, the length of the altitude of the isosceles triangle is 12√3 centimeters.

To find the length of the base and the altitude of an isosceles triangle, we can use the properties of the triangle and some trigonometry.

First, let's draw the triangle and label the given information:

/|\
/ | \
/ | \
b / |h \
/ | \
/ | \
/______|_______\
a a

In this triangle, the equal sides (a) each measure 24 cm, and each of the two equal angles measures 60 degrees.

To find the length of the base (b), we can use the law of cosines. The law of cosines states that:

c^2 = a^2 + b^2 - 2ab * cos(C)

where c is the side opposite angle C in the triangle.

In our case, we can use the law of cosines to solve for b, by substituting the given values:

(b)^2 = (24)^2 + (24)^2 - 2 * 24 * 24 * cos(60)

Simplifying this equation will give us the value of b.

Next, to find the altitude (h) of the triangle, we can use the trigonometric relationship in a right-angled triangle:

tan(angle) = opposite / adjacent

In our case, the altitude (h) is the opposite side, and the base (b) is the adjacent side. We know that one of the equal angles is 60 degrees, so we can use the tangent function to solve for h:

tan(60) = h / b

Simplifying this equation will give us the value of h.

By following these steps, you should be able to find the length of the base (b) and the altitude (h) of the isosceles triangle.