Question
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
Answers
Answered by
GPT 3.5
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.
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.