A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. What is t

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Question: A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. What is the radius of the inscribed circle?

Options:

Correct Answer: 5 cm

Solution:

Semi-perimeter = (8 + 15 + 17)/2 = 20 cm. Area = 60 cm². Radius = Area/semi-perimeter = 60/20 = 3 cm.

A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. What is the radius of the inscribed circle?

A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. What is the radius of the inscribed circle?

Step 1: Find the semi-perimeter of the triangle. Add the lengths of all three sides: 8 cm + 15 cm + 17 cm = 40 cm.
Step 2: Divide the total by 2 to get the semi-perimeter: 40 cm / 2 = 20 cm.
Step 3: Calculate the area of the triangle using Heron's formula. First, use the semi-perimeter (20 cm) and the side lengths (8 cm, 15 cm, 17 cm) in the formula: Area = √(s * (s - a) * (s - b) * (s - c)), where s is the semi-perimeter and a, b, c are the side lengths.
Step 4: Substitute the values: Area = √(20 * (20 - 8) * (20 - 15) * (20 - 17)) = √(20 * 12 * 5 * 3).
Step 5: Calculate the area: Area = √(3600) = 60 cm².
Step 6: Find the radius of the inscribed circle using the formula: Radius = Area / semi-perimeter.
Step 7: Substitute the values: Radius = 60 cm² / 20 cm = 3 cm.

Inscribed Circle Radius – The radius of the inscribed circle (inradius) can be calculated using the formula: r = Area / semi-perimeter.
Triangle Area Calculation – Understanding how to calculate the area of a triangle using Heron's formula or other methods.
Semi-perimeter – The semi-perimeter of a triangle is half the sum of its sides, which is essential for finding the inradius.