A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm,

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Question: A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm, 8 cm, and 9 cm, what is the radius of the inscribed circle?

Options:

3 cm
4 cm
5 cm
6 cm

Correct Answer: 4 cm

Solution:

The area A of the triangle can be calculated using Heron\'s formula. The semi-perimeter s = (7 + 8 + 9) / 2 = 12. The area A = √(s(s-a)(s-b)(s-c)) = √(12(12-7)(12-8)(12-9)) = √(12*5*4*3) = 12√5. The radius r = A/s = (12√5)/12 = √5 cm, which is approximately 4 cm.

A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm,

Practice Questions

A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm, 8 cm, and 9 cm, what is the radius of the inscribed circle?

3 cm
4 cm
5 cm
6 cm

Questions & Step-by-Step Solutions

A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm, 8 cm, and 9 cm, what is the radius of the inscribed circle?

Step 1: Identify the lengths of the sides of the triangle. They are 7 cm, 8 cm, and 9 cm.
Step 2: Calculate the semi-perimeter (s) of the triangle using the formula s = (a + b + c) / 2, where a, b, and c are the side lengths. Here, s = (7 + 8 + 9) / 2 = 12 cm.
Step 3: Use Heron's formula to find the area (A) of the triangle. The formula is A = √(s(s-a)(s-b)(s-c).
Step 4: Calculate (s-a), (s-b), and (s-c): (s-7) = (12-7) = 5, (s-8) = (12-8) = 4, (s-9) = (12-9) = 3.
Step 5: Substitute the values into Heron's formula: A = √(12 * 5 * 4 * 3).
Step 6: Calculate the product inside the square root: 12 * 5 = 60, then 60 * 4 = 240, and finally 240 * 3 = 720.
Step 7: Find the square root of 720: A = √720 = 12√5 cm² (this is the area of the triangle).
Step 8: Calculate the radius (r) of the inscribed circle using the formula r = A / s.
Step 9: Substitute the area and semi-perimeter into the formula: r = (12√5) / 12.
Step 10: Simplify the expression: r = √5 cm.

Heron's Formula – A method to calculate the area of a triangle when the lengths of all three sides are known.
Inscribed Circle Radius – The radius of the circle that can be inscribed within a triangle, calculated using the area and semi-perimeter.
Semi-perimeter – Half of the perimeter of the triangle, used in Heron's formula to find the area.

A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm,

What’s inside this PDF?

A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm,

Practice Questions

Questions & Step-by-Step Solutions

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