An isosceles triangle has two 10.0-inch sides and a 2w-inch side. Find the radius of the inscribed circle of this triangle, in the cases w = 5.00, w = 6.00, and w = 8.00.
To find the radius of the inscribed circle of an isosceles triangle, we can use the formula:
r = (a * sin(B/2)) / (1+sin(B/2))
Where:
r = radius of the inscribed circle
a = length of one of the equal sides of the triangle
B = vertex angle of the triangle
In this case, we have an isosceles triangle with two 10.0-inch sides and a 2w-inch side. Since it is isosceles, two sides are equal.
Let's first calculate the vertex angle B:
B = arccos((a^2 + a^2 - (2w)^2) / (2 * a * a))
B = arccos((10.0^2 + 10.0^2 - (2w)^2) / (2 * 10.0 * 10.0))
Now, we can substitute the value of B into the formula for r, with different values of w:
For w = 5.00:
B = arccos((10.0^2 + 10.0^2 - (2 * 5.00)^2) / (2 * 10.0 * 10.0))
r = (10.0 * sin(B/2)) / (1+sin(B/2))
For w = 6.00:
B = arccos((10.0^2 + 10.0^2 - (2 * 6.00)^2) / (2 * 10.0 * 10.0))
r = (10.0 * sin(B/2)) / (1+sin(B/2))
For w = 8.00:
B = arccos((10.0^2 + 10.0^2 - (2 * 8.00)^2) / (2 * 10.0 * 10.0))
r = (10.0 * sin(B/2)) / (1+sin(B/2))
By substituting the values of B into the formula for r, we can find the radius of the inscribed circle in each case.