A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8,

A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?

A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?

Step 1: Identify the lengths of the sides of the triangle. They are 7, 8, and 9 units.
Step 2: Calculate the semi-perimeter of the triangle. The semi-perimeter (s) is found by adding the lengths of the sides and dividing by 2: s = (7 + 8 + 9) / 2 = 12 units.
Step 3: Use Heron's formula to find the area of the triangle. First, calculate the semi-perimeter (s) which we already found to be 12 units.
Step 4: Apply Heron's formula: Area = √(s * (s - a) * (s - b) * (s - c)), where a, b, and c are the side lengths. Here, Area = √(12 * (12 - 7) * (12 - 8) * (12 - 9)).
Step 5: Calculate the values inside the square root: Area = √(12 * 5 * 4 * 3).
Step 6: Simplify the multiplication: Area = √(720).
Step 7: Calculate the square root: Area = 24 square units.
Step 8: Now, find the radius of the inscribed circle (r) using the formula r = Area / semi-perimeter.
Step 9: Substitute the values: r = 24 / 12.
Step 10: Calculate the radius: r = 2 units.

Heron's Formula – A method to calculate the area of a triangle when the lengths of all three sides are known.
Inscribed Circle Radius – The radius of the circle that can be inscribed within a triangle, calculated using the area and semi-perimeter.
Semi-perimeter – Half of the perimeter of the triangle, used in the calculation of the inscribed circle's radius.