A circle is inscribed in a triangle. If the sides of the triangle are 7 cm, 8 cm

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Question: A circle is inscribed in a triangle. If the sides of the triangle are 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 radius r of the inscribed circle can be found using the formula r = A/s, where A is the area and s is the semi-perimeter. The semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. The area A can be calculated using Heron\'s formula: A = √[s(s-a)(s-b)(s-c)] = √[12(12-7)(12-8)(12-9)] = √[12*5*4*3] = √720 = 12√5. Thus, r = A/s = (12√5)/12 = √5 cm, which is approximately 2.24 cm.

A circle is inscribed in a triangle. If the sides of the triangle are 7 cm, 8 cm

Practice Questions

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

Step 1: Identify the sides of the triangle. The sides 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. 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: Substitute the values into Heron's formula. A = √[12(12-7)(12-8)(12-9)] = √[12 * 5 * 4 * 3].
Step 5: Calculate the product inside the square root: 12 * 5 = 60, 60 * 4 = 240, 240 * 3 = 720. So, A = √720.
Step 6: Simplify √720. This can be written as √(144 * 5) = 12√5.
Step 7: Now, use the formula for the radius (r) of the inscribed circle: r = A / s. Substitute A = 12√5 and s = 12.
Step 8: Calculate r: r = (12√5) / 12 = √5 cm.
Step 9: Approximate √5 to find the numerical value of the radius. √5 is approximately 2.24 cm.

Inscribed Circle Radius – Understanding how to calculate the radius of a circle inscribed in a triangle using the area and semi-perimeter.
Heron's Formula – Applying Heron's formula to find the area of a triangle when the lengths of all three sides are known.
Semi-perimeter – Calculating the semi-perimeter of a triangle as half the sum of its sides.

A circle is inscribed in a triangle. If the sides of the triangle are 7 cm, 8 cm

What’s inside this PDF?

A circle is inscribed in a triangle. If the sides of the triangle are 7 cm, 8 cm

Practice Questions

Questions & Step-by-Step Solutions

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