Question

If $$\text{a = 13,b = 12,c = 5}$$in $$△\text{ABC}$$where a, b, c are the sides of a triangle, then the value of
$$\text{sin}{\displaystyle \frac{A}{2}}=$$
$$\begin{array}{c}\left(a\right){\displaystyle \frac{1}{\sqrt{5}}}\\ \left(b\right){\displaystyle \frac{2}{3}}\\ \left(c\right)\sqrt{{\displaystyle \frac{32}{35}}}\\ \left(d\right){\displaystyle \frac{1}{\sqrt{2}}}\end{array}$$

Answer 1

Hint- Try to figure out whether the triangular sides formulated above forms a right angled triangle or not.

In the above figure we can see that $△\text{ABC}$has the sides $\text{a = 13,b = 12 and c = 5}$
Now if we try to apply Pythagoras theorem which states that if $\text{hypotenuse}{\text{s}}^2=\text{perpendicular}{\text{r}}^2+\text{bas}{\text{e}}^2$then the triangle will be a right angle triangle.
Thus clearly ${\text{a}}^2={\text{b}}^2+{\text{c}}^2$that is $\text{1}{\text{3}}^2={12}^2+5^2$or $\text{169 = 144 + 25}$
Hence we can say that the above triangle is a right angle triangle and from the above figure it is clear that it is right angled at A that is$\mathrm{\angle }\text{A = 90}$.
Let’s discuss why it is right angled at A only and not B or C?
Because $\mathrm{\angle }\text{A}$is the opposite angle to the greatest side of the triangle and Pythagoras theorem is also applicable.
Hence $\text{sin}{\displaystyle \frac{A}{2}}=\sin {\displaystyle \frac{90}{2}}=\sin 45={\displaystyle \frac{1}{\sqrt{2}}}$
Hence option (d) is the correct option.
Note-If a triangle is found obeying the Pythagoras theorem then the angle which is always opposite to the greatest side is 90 degree or in other words if a triangle is obeying Pythagoras theorem than it is right angled at the angle which is exactly opposite to the greatest side in that triangle.