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

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

Solution: Using the formula for the radius of the incircle r = A/s, where A is the area and s is the semi-perimeter.

Steps: 14

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 side lengths 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: Substitute these values: A = √(12 * 5 * 4 * 3).
Step 9: Calculate the product: 12 * 5 = 60, then 60 * 4 = 240, and finally 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.