The figure below is a triangle ABC inscribed in a circle.AB=6cm,BC=9cm and angle ABC=80 degrees.Calculate the radius of the circle
To find the radius of the circle inscribed in triangle ABC, we can use the formula for the radius of the inscribed circle in a triangle given the side lengths and the angles.
The formula for the radius of the inscribed circle (r) is:
r = (a + b - c) / 2
where a, b, and c are the side lengths of the triangle.
In this case, we have AB = 6 cm, BC = 9 cm, and AC (or c) is the unknown side length.
To find AC, we can use the Law of Cosines:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(ABC)
AC^2 = 6^2 + 9^2 - 2 * 6 * 9 * cos(80)
AC^2 = 36 + 81 - 108 * cos(80)
AC^2 = 36 + 81 - 108 * (-0.17364817766693033)
AC^2 = 36 + 81 + 18.808002806603715
AC^2 = 135.80800280660372
AC ≈ 11.64 cm
Now that we know AC, we can calculate the radius of the inscribed circle:
r = (AB + BC - AC) / 2
r = (6 + 9 - 11.64) / 2
r = 3.36 / 2
r = 1.68 cm
Therefore, the radius of the circle is approximately 1.68 cm.
To calculate the radius of the circle inscribed in triangle ABC, we need to use the following formula:
r = (a * b * c) / (4 * A)
where r is the radius of the circle, a, b, and c are the lengths of the sides of the triangle, and A is the area of the triangle.
First, let's calculate the area of triangle ABC using Heron's Formula:
s = (a + b + c) / 2
A = √(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter of the triangle.
s = (6 + 9 + 6) / 2 = 21 / 2 = 10.5
A = √(10.5 * (10.5 - 6) * (10.5 - 9) * (10.5 - 6))
A = √(10.5 * 4.5 * 1.5 * 4.5)
A = √(170.34375)
A ≈ 13.05
Now, let's substitute the values into the formula to calculate the radius:
r = (6 * 9 * 6) / (4 * 13.05)
r = 54 / 52.2
r ≈ 1.04 cm
Therefore, the radius of the circle inscribed in triangle ABC is approximately 1.04 cm.