A circle is inscribed in a triangle with sides 6 cm, 8 cm, and 10 cm. What is th

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Question: A circle is inscribed in a triangle with sides 6 cm, 8 cm, and 10 cm. What is the radius of the inscribed circle?

Options:

Correct Answer: 3 cm

Solution:

Semi-perimeter = (6+8+10)/2 = 12 cm; Area = √(12(12-6)(12-8)(12-10)) = 24 cm²; Radius = Area/semi-perimeter = 24/12 = 2 cm.

A circle is inscribed in a triangle with sides 6 cm, 8 cm, and 10 cm. What is the radius of the inscribed circle?

A circle is inscribed in a triangle with sides 6 cm, 8 cm, and 10 cm. What is the radius of the inscribed circle?

Step 1: Find the semi-perimeter of the triangle. Add the lengths of the sides: 6 cm + 8 cm + 10 cm = 24 cm. Then divide by 2: 24 cm / 2 = 12 cm.
Step 2: Use Heron's formula to find the area of the triangle. The formula is Area = √(s(s-a)(s-b)(s-c), where s is the semi-perimeter and a, b, c are the sides of the triangle. Here, s = 12 cm, a = 6 cm, b = 8 cm, c = 10 cm.
Step 3: Calculate (s-a), (s-b), and (s-c): (12-6) = 6, (12-8) = 4, (12-10) = 2.
Step 4: Substitute these values into the area formula: Area = √(12 * 6 * 4 * 2).
Step 5: Calculate the product: 12 * 6 = 72, then 72 * 4 = 288, and finally 288 * 2 = 576.
Step 6: Find the square root of 576: √576 = 24 cm². So, the area of the triangle is 24 cm².
Step 7: Now, find the radius of the inscribed circle using the formula: Radius = Area / semi-perimeter.
Step 8: Substitute the values: Radius = 24 cm² / 12 cm = 2 cm.

Inscribed Circle Radius – The radius of the inscribed circle (inradius) can be calculated using the formula: r = Area / semi-perimeter.
Heron's Formula – Heron's formula is used to calculate the area of a triangle when the lengths of all three sides are known.
Semi-perimeter – The semi-perimeter of a triangle is half the sum of its sides, which is essential for calculating the area and the inradius.