Answer

Verified

471k+ views

Hint-Using the vertices first find out the length of the sides of the triangle and proceed

Let us consider the three vertices to be A=(1,$\sqrt 3 $ ),B=(0,0),C=(2,0) which in turn forms a $\vartriangle ABC$

So, to find out the length make use of the distance formula and solve it

$d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{({y_2} - {y_1})}^2}} $

So, we will get the length of the side AB=$\sqrt {{{\left( {0 - 1} \right)}^2} + {{\left( {0 - \sqrt 3 } \right)}^2}} = \sqrt {1 + 3} = \sqrt 4 = 2$ =c

The length of the side CA=$\sqrt {{{\left( {2 - 1} \right)}^2} + {{\left( {0 - \sqrt 3 } \right)}^2}} = \sqrt {1 + 3} = \sqrt 4 = 2$=b

The length of the side BC=$\sqrt {{{\left( {2 - 0} \right)}^2} + {{\left( {0 - 0} \right)}^2}} = \sqrt {4 + 0} = \sqrt 4 = 2$ =a

So, from this , since the length of sides AB=BC=CA=2, we can conclude that it is an equilateral triangle

Since it is an equilateral triangle, the incentre is nothing but equal to the centroid

So, we can write the coordinates of the incentre are

$\begin{gathered}

\left( {\frac{{a{x_1} + b{x_2} + c{x_3}}}{{a + b + c}},\frac{{a{y_1} + b{y_2} + c{y_3}}}{{a + b + c}}} \right) = \left( {\frac{{2 \times 1 + 2 \times 0 + 2 \times 2}}{{2 + 2 + 2}},\frac{{2 \times \sqrt 3 + 2 \times 0 + 2 \times 0}}{{2 + 2 + 2}}} \right) \\

= \left( {\frac{6}{6},\frac{{2\sqrt 3 }}{6}} \right) = \left( {1,\frac{1}{{\sqrt 3 }}} \right) \\

\end{gathered} $

So, option D is the correct answer to this question.

Note: Whenever we are given with these kind of questions, first find out what type of

triangle is formed from these sides and then apply the formula with respect to the type of

triangle formed and solve

Let us consider the three vertices to be A=(1,$\sqrt 3 $ ),B=(0,0),C=(2,0) which in turn forms a $\vartriangle ABC$

So, to find out the length make use of the distance formula and solve it

$d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{({y_2} - {y_1})}^2}} $

So, we will get the length of the side AB=$\sqrt {{{\left( {0 - 1} \right)}^2} + {{\left( {0 - \sqrt 3 } \right)}^2}} = \sqrt {1 + 3} = \sqrt 4 = 2$ =c

The length of the side CA=$\sqrt {{{\left( {2 - 1} \right)}^2} + {{\left( {0 - \sqrt 3 } \right)}^2}} = \sqrt {1 + 3} = \sqrt 4 = 2$=b

The length of the side BC=$\sqrt {{{\left( {2 - 0} \right)}^2} + {{\left( {0 - 0} \right)}^2}} = \sqrt {4 + 0} = \sqrt 4 = 2$ =a

So, from this , since the length of sides AB=BC=CA=2, we can conclude that it is an equilateral triangle

Since it is an equilateral triangle, the incentre is nothing but equal to the centroid

So, we can write the coordinates of the incentre are

$\begin{gathered}

\left( {\frac{{a{x_1} + b{x_2} + c{x_3}}}{{a + b + c}},\frac{{a{y_1} + b{y_2} + c{y_3}}}{{a + b + c}}} \right) = \left( {\frac{{2 \times 1 + 2 \times 0 + 2 \times 2}}{{2 + 2 + 2}},\frac{{2 \times \sqrt 3 + 2 \times 0 + 2 \times 0}}{{2 + 2 + 2}}} \right) \\

= \left( {\frac{6}{6},\frac{{2\sqrt 3 }}{6}} \right) = \left( {1,\frac{1}{{\sqrt 3 }}} \right) \\

\end{gathered} $

So, option D is the correct answer to this question.

Note: Whenever we are given with these kind of questions, first find out what type of

triangle is formed from these sides and then apply the formula with respect to the type of

triangle formed and solve

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

A group of fish is known as class 7 english CBSE

The highest dam in India is A Bhakra dam B Tehri dam class 10 social science CBSE

Write all prime numbers between 80 and 100 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Onam is the main festival of which state A Karnataka class 7 social science CBSE

Who administers the oath of office to the President class 10 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Kolkata port is situated on the banks of river A Ganga class 9 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE