Answer:
To find the radius of the circle inscribed in an isosceles triangle, we can use the following formula:
r = (a/2) * cot(π/n)
where r is the radius of the inscribed circle, a is the length of one of the equal sides of the isosceles triangle, and n is the number of sides of the polygon inscribed in the circle.
In this case, we have an isosceles triangle with two sides of 5cm and one side of 6cm. Since the triangle is isosceles, the angle opposite the 6cm side is bisected by the altitude and therefore, the two smaller angles are congruent. Let x be the measure of one of these angles. Using the Law of Cosines, we can solve for x:
6^2 = 5^2 + 5^2 - 2(5)(5)cos(x)
36 = 50 - 50cos(x)
cos(x) = (50 - 36)/50
cos(x) = 0.28
x = cos^-1(0.28) ≈ 73.7°
Since the isosceles triangle has two equal sides of length 5cm, we can divide the triangle into two congruent right triangles by drawing an altitude from the vertex opposite the 6cm side to the midpoint of the 6cm side. The length of this altitude can be found using the Pythagorean theorem:
(5/2)^2 + h^2 = 5^2
25/4 + h^2 = 25
h^2 = 75/4
h = sqrt(75)/2 = (5/2)sqrt(3)
Now we can find the radius of the inscribed circle using the formula:
r = (a/2) * cot(π/n)
where a = 5cm and n = 3 (since the circle is inscribed in a triangle, which is a 3-sided polygon). We can also use the fact that the distance from the center of the circle to the midpoint of each side of the triangle is equal to the radius of the circle. Therefore, the radius of the circle is equal to the altitude of the triangle from the vertex opposite the 6cm side:
r = (5/2) * cot(π/3) = (5/2) * sqrt(3) ≈ 2.89 cm
Therefore, the radius of the circle inscribed in the isosceles triangle with sides 5cm, 5cm, and 6cm is approximately 2.89 cm.