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

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

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

Step 1: Calculate the semi-perimeter (s) of the triangle. The semi-perimeter is found using the formula s = (a + b + c) / 2, where a, b, and c are the sides of the triangle.
Step 2: Substitute the values of the sides into the formula: s = (7 + 8 + 9) / 2.
Step 3: Calculate the sum of the sides: 7 + 8 + 9 = 24.
Step 4: Divide the sum by 2 to find the semi-perimeter: s = 24 / 2 = 12.
Step 5: Now, calculate the area (A) of the triangle using Heron's formula: A = √(s * (s - a) * (s - b) * (s - c)).
Step 6: Substitute the values into Heron's formula: A = √(12 * (12 - 7) * (12 - 8) * (12 - 9)).
Step 7: Calculate the differences: (12 - 7) = 5, (12 - 8) = 4, (12 - 9) = 3.
Step 8: Now substitute these values: A = √(12 * 5 * 4 * 3).
Step 9: Calculate the product: 12 * 5 = 60, 60 * 4 = 240, 240 * 3 = 720.
Step 10: Find the square root: A = √720.
Step 11: Simplify √720 to find the area: A = 12√5.
Step 12: Now, use the formula for the radius of the incircle: r = A / s.
Step 13: Substitute the area and semi-perimeter into the formula: r = (12√5) / 12.
Step 14: Simplify the expression: r = √5.

Incircle Radius Calculation – The radius of the incircle of a triangle can be calculated using the formula r = A/s, where A is the area of the triangle and s is the semi-perimeter.
Area of Triangle – To find the area A of a triangle with given sides, Heron's formula can be used, which requires calculating the semi-perimeter first.
Semi-perimeter – The semi-perimeter s of a triangle is calculated as s = (a + b + c) / 2, where a, b, and c are the lengths of the sides.