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\[{\text{sin}}\dfrac{A}{2} = \]

\[\begin{gathered}

\left( a \right){\text{ }}\dfrac{1}{{\sqrt 5 }} \\

\left( b \right){\text{ }}\dfrac{2}{3} \\

\left( c \right){\text{ }}\sqrt {\dfrac{{32}}{{35}}} \\

\left( d \right){\text{ }}\dfrac{1}{{\sqrt 2 }} \\

\end{gathered} \]

Answer

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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 $\vartriangle {\text{ABC}}$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$\angle {\text{A = 90}}$.

Let’s discuss why it is right angled at A only and not B or C?

Because $\angle {\text{A}}$is the opposite angle to the greatest side of the triangle and Pythagoras theorem is also applicable.

Hence ${\text{sin}}\dfrac{A}{2} = \sin \dfrac{{90}}{2} = \sin 45 = \dfrac{1}{{\sqrt 2 }}$

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.

In the above figure we can see that $\vartriangle {\text{ABC}}$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$\angle {\text{A = 90}}$.

Let’s discuss why it is right angled at A only and not B or C?

Because $\angle {\text{A}}$is the opposite angle to the greatest side of the triangle and Pythagoras theorem is also applicable.

Hence ${\text{sin}}\dfrac{A}{2} = \sin \dfrac{{90}}{2} = \sin 45 = \dfrac{1}{{\sqrt 2 }}$

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.