# If \[{\text{a = 13,b = 12,c = 5}}\]in \[\vartriangle {\text{ABC}}\]where a, b, c are the sides of a triangle, then the value of

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

\[\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} \]

Last updated date: 15th Mar 2023

•

Total views: 307.5k

•

Views today: 8.87k

Answer

Verified

307.5k+ views

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.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE