A circle is inscribed in a triangle with sides 7 cm, 8 cm, and 9 cm. What is the

A circle is inscribed in a triangle with sides 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?

A circle is inscribed in a triangle with sides 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?

Step 1: Find the semi-perimeter of the triangle. Add the lengths of the sides: 7 cm + 8 cm + 9 cm = 24 cm.
Step 2: Divide the total by 2 to get the semi-perimeter: 24 cm / 2 = 12 cm.
Step 3: Use the semi-perimeter to find the area of the triangle using Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter and a, b, c are the sides of the triangle.
Step 4: Substitute the values into the formula: Area = √[12(12-7)(12-8)(12-9)] = √[12 * 5 * 4 * 3].
Step 5: Calculate the product inside the square root: 12 * 5 = 60, 60 * 4 = 240, 240 * 3 = 720.
Step 6: Find the square root of 720: Area = √720 = 12√5 cm².
Step 7: Now, find the radius of the inscribed circle using the formula: Radius = Area / semi-perimeter.
Step 8: Substitute the area and semi-perimeter: Radius = (12√5) / 12.
Step 9: Simplify the expression: Radius = √5 cm.

Inscribed Circle Radius – The radius of the inscribed circle (inradius) can be calculated using the formula: r = Area / semi-perimeter.
Heron's Formula – The area of a triangle can be calculated using Heron's formula, which requires the semi-perimeter and the lengths of the sides.
Semi-perimeter Calculation – The semi-perimeter is half the sum of the triangle's sides, which is essential for both area and radius calculations.