The lenght of each of the two sides of an isisceles triangle is 10 meters. The angle between the two congruent sides is x. find the area of the triangle as a function of x/2.
First draw a picture. Split the isosceles triangle down the middle of triangle, bisecting the angle x. Now you have divided the isosceles triangle into two right angled triangles, each with area of 1/2 height * base.
So the area of the isosceles is the sum of both of these smaller right angled triangles, or height * base
The base of the right angle triangle is given by base = 10 * sin(x/2)
The height of the right angle triangle is given by height = 10*cos(x/2)
area = 100*cos(x/2)*sin(x/2)
To find the area of an isosceles triangle, we need to know either the height or the base length. In this case, we have the length of the two congruent sides (10 meters), but we also need to determine the length of the base or the height.
Let's consider the length of the base. Since an isosceles triangle has two congruent sides, the base is the third side of the triangle. To find the length of the base, we can use the Law of Cosines.
The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we have an isosceles triangle with two sides of length 10, so let's use the length of the base as b, and the angle between the congruent sides as C.
Using the Law of Cosines, we have:
b^2 = 10^2 + 10^2 - 2 * 10 * 10 * cos(x)
Simplifying this equation will give us the length of the base (b) in terms of x.
Next, we need to find the height of the triangle. Since the triangle is isosceles, the height is also the perpendicular bisector of the base, cutting it into two equal halves. So, the height will create two right triangles.
Let's call the length of the height h. Using the Pythagorean theorem on one of the right triangles, we have:
h^2 = (b/2)^2 - 10^2
Simplifying this equation will give us the length of the height (h) in terms of x.
Now that we have the base length (b) and the height (h) in terms of x, we can calculate the area of the triangle using the formula:
Area = (1/2) * base * height
Substituting the expressions for the base and height, we obtain the area of the triangle as a function of x/2.