Question

In a $\Delta ABC$, if $a=13, b=14$ and $c=15$ then the value of $\tan \left(\frac{A}{2}\right)$ is

• A. $\frac{1}{4}$
• B. $\frac{3}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{6}$

Hint:

Alternatively, you can use the formula $\tan \frac{A}{2} = \frac{(s-b)(s-c)}{\Delta}$ which is often faster if you have already calculated the area. Here: $(7 \times 6) / 84 = 42/84 = 1/2$.

Answer 1

The Correct Option is C

To solve for the half-angle tangent, we first need to calculate the semi-perimeter ($s$) and the area of the triangle ($\Delta$). Step 1: Calculate Semi-perimeter ($s$) $$s = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\lt strong\gt Step 2: Calculate the Area ($\Delta$) using Heron's Formula\lt /strong\gt \Delta = \sqrt{s(s-a)(s-b)(s-c)}$$ $$\Delta = \sqrt{21(21-13)(21-14)(21-15)}$$ $$\Delta = \sqrt{21 \times 8 \times 7 \times 6}$$ $$\Delta = \sqrt{7056} = 84$$

Step 3: Apply the Half-Angle Formula for Tangent The formula for $\tan \frac{A}{2}$ is: $$\tan \frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$$ Substituting the values: $$\tan \frac{A}{2} = \sqrt{\frac{(21-14)(21-15)}{21(21-13)}}$$ $$\tan \frac{A}{2} = \sqrt{\frac{7 \times 6}{21 \times 8}}$$ $$\tan \frac{A}{2} = \sqrt{\frac{42}{168}}$$ $$\tan \frac{A}{2} = \sqrt{\frac{1}{4}} = \frac{1}{2}$$ Thus, the value of $\tan \frac{A}{2}$ is $1/2$.