Question

In a triangle ABC, if \( a = 13, b = 14, c = 15 \), then \( r_1 = \)

• A. \( \frac{23}{2} \)
• B. \( \frac{21}{2} \)
• C. \( \frac{25}{2} \)
• D. \( \frac{26}{3} \)

Hint:

Use Heron’s formula to calculate the area of the triangle when the sides are given, and use the formula for the inradius to find the desired value.

Answer 1

The Correct Option is B

Step 1: Calculate the semi-perimeter \( s \).
The semi-perimeter \( s \) is given by the formula: \[ s = \frac{a + b + c}{2} \] Substituting the values of the sides \( a = 13 \), \( b = 14 \), and \( c = 15 \): \[ s = \frac{13 + 14 + 15}{2} = 21 \] Step 2: Calculate the area \( A \) using Heron's formula.
Heron’s formula for the area \( A \) is: \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] Substituting the values: \[ A = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \times 8 \times 7 \times 6} \] \[ A = \sqrt{7056} = 84 \] Step 3: Calculate the inradius \( r_1 \).
The inradius \( r_1 \) is given by the formula: \[ r_1 = \frac{A}{s} \] Substituting the values of \( A = 84 \) and \( s = 21 \): \[ r_1 = \frac{84}{21} = 4 \] Conclusion:
Thus, the correct value of the inradius \( r_1 \) is \(\frac{21}{2}\).