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What is the circumradius of a triangle with sides 6, 8, and 10?

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Question: What is the circumradius of a triangle with sides 6, 8, and 10?

Options:

  1. 5
  2. 6
  3. 7
  4. 8

Correct Answer: 5

Solution: 

Circumradius R = (abc)/(4K), where K is the area. Area K = 24 (using Heron\'s formula). R = (6*8*10)/(4*24) = 5.

What is the circumradius of a triangle with sides 6, 8, and 10?

Practice Questions

Q1
What is the circumradius of a triangle with sides 6, 8, and 10?
  1. 5
  2. 6
  3. 7
  4. 8

Questions & Step-by-Step Solutions

What is the circumradius of a triangle with sides 6, 8, and 10?
Correct Answer: 5
  • Step 1: Identify the sides of the triangle. The sides are 6, 8, and 10.
  • Step 2: Calculate the semi-perimeter (s) of the triangle using the formula s = (a + b + c) / 2. Here, s = (6 + 8 + 10) / 2 = 12.
  • Step 3: Use Heron's formula to find the area (K) of the triangle. K = √(s * (s - a) * (s - b) * (s - c)).
  • Step 4: Substitute the values into Heron's formula: K = √(12 * (12 - 6) * (12 - 8) * (12 - 10)) = √(12 * 6 * 4 * 2).
  • Step 5: Calculate the area: K = √(576) = 24.
  • Step 6: Use the circumradius formula R = (abc) / (4K). Here, a = 6, b = 8, c = 10, and K = 24.
  • Step 7: Substitute the values into the circumradius formula: R = (6 * 8 * 10) / (4 * 24).
  • Step 8: Calculate the numerator: 6 * 8 * 10 = 480.
  • Step 9: Calculate the denominator: 4 * 24 = 96.
  • Step 10: Divide the numerator by the denominator: R = 480 / 96 = 5.
  • Circumradius of a Triangle – The circumradius is the radius of the circumcircle that passes through all three vertices of the triangle. It can be calculated using the formula R = (abc)/(4K), where a, b, and c are the lengths of the sides and K is the area of the triangle.
  • Heron's Formula – Heron's formula is used to calculate the area of a triangle when the lengths of all three sides are known. It states that K = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter of the triangle.
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