Question

The circumradius of a triangle whose sides are 10 units, 8 units and 6 units is ______.

• A. 4 units
• B. 2 units
• C. 3 units
• D. 5 units

Hint:

While the general formula for the circumradius of any triangle is $R = \frac{abc}{4\Delta}$ (where $\Delta$ is the area), spotting a Pythagorean triple instantly reduces the calculation time to 3 seconds.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the three side lengths of a triangle ($a=6, b=8, c=10$) and need to find its circumradius ($R$), which is the radius of the circle that perfectly passes through all three of its vertices.

Step 2: Key Formula or Approach:
Always check if given side lengths form a Pythagorean triple. If $a^2 + b^2 = c^2$, the triangle is right-angled.
A fundamental property of right-angled triangles inscribed in a circle is that the hypotenuse serves as the exact diameter of the circumcircle.
Therefore, the circumradius $R = \frac{\text{Hypotenuse}}{2}$.

Step 3: Detailed Explanation:
Let's check the squares of the sides:
$6^2 = 36$
$8^2 = 64$
$10^2 = 100$
Since $36 + 64 = 100$, the sides perfectly satisfy the Pythagorean theorem.
Thus, this is a right-angled triangle, and the longest side (10 units) is the hypotenuse.
$$R = \frac{\text{Hypotenuse}}{2} = \frac{10}{2} = 5 \text{ units}$$

Step 4: Final Answer:
The circumradius is 5 units, matching option (d).