What is the circumradius R of a triangle with sides a = 7, b = 24, c = 25?

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Question: What is the circumradius R of a triangle with sides a = 7, b = 24, c = 25?

Options:

Correct Answer: 12

Solution:

R = (abc) / (4 * Area). Area = 84, R = (7 * 24 * 25) / (4 * 84) = 12.

What is the circumradius R of a triangle with sides a = 7, b = 24, c = 25?

What is the circumradius R of a triangle with sides a = 7, b = 24, c = 25?

Step 1: Identify the sides of the triangle. Here, a = 7, b = 24, and c = 25.
Step 2: Use the formula for the circumradius R of a triangle: R = (abc) / (4 * Area).
Step 3: Calculate the area of the triangle using Heron's formula. First, find the semi-perimeter s = (a + b + c) / 2 = (7 + 24 + 25) / 2 = 28.
Step 4: Calculate the area using Heron's formula: Area = sqrt(s * (s - a) * (s - b) * (s - c)).
Step 5: Substitute the values: Area = sqrt(28 * (28 - 7) * (28 - 24) * (28 - 25)) = sqrt(28 * 21 * 4 * 3).
Step 6: Calculate the area: Area = sqrt(28 * 21 * 12) = sqrt(7056) = 84.
Step 7: Now substitute the area back into the circumradius formula: R = (7 * 24 * 25) / (4 * 84).
Step 8: Calculate the numerator: 7 * 24 * 25 = 4200.
Step 9: Calculate the denominator: 4 * 84 = 336.
Step 10: Divide the numerator by the denominator: R = 4200 / 336 = 12.

Circumradius of a Triangle – The circumradius R of a triangle can be calculated using the formula R = (abc) / (4 * Area), where a, b, and c are the lengths of the sides and Area is the area of the triangle.
Area of a Triangle – The area of a triangle can be calculated using Heron's formula or by using the base-height method, which is essential for finding the circumradius.