Answer

Verified

450k+ views

Hint: A right angled triangle can always be inscribed inside a circle with the hypotenuse as the diameter of the circle and the midpoint of the hypotenuse of the triangle will be the circumcentre of the triangle .

Consider the figure alongside . In the question it is given that $B$ is the orthocentre. We also know that $B$ is one of the vertices of the triangle $ABC$.

So , we can conclude that $ABC$ is a right-angled triangle.

Now , we know that a right angled triangle can be inscribed in a circle with the hypotenuse as the diameter.

So , we can conclude that $AC$ is the diameter of a circle.

So , the centre of this circle will be the midpoint of the diameter.

Now, we know the midpoint of line joining the points $\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right)$ is given as:

$\left( \dfrac{\left( {{x}_{1}}+{{x}_{2}} \right)}{2},\dfrac{\left( {{y}_{1}}+{{y}_{2}} \right)}{2} \right)...............\left( i \right)$

Now , we will consider $\left( x,y \right)$ to be the coordinates of $C$.

We know , circumcentre of a triangle is the centre of the circle circumscribing the triangle.

So , the circumcentre $D\left( 2,4 \right)$ is the midpoint of the line joining $A\left( 0,0 \right)$ and $C\left( x,y \right)$.

Now , we will use the equation $\left( i \right)$ to find the coordinates of the vertex $C$.

So , from equation$\left( i \right)$ , we have

$2=\dfrac{x+0}{2}\Rightarrow x=4$

And $\text{ 4}=\dfrac{y+0}{2}\Rightarrow y=8$

So , the coordinates of vertex $C$ are $\left( 4,8 \right)$.

Option B - $\left( 4,8 \right)$ is correct answer

Note: The midpoint of line joining the points $\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right)$ is given as:

$\left( \dfrac{\left( {{x}_{1}}+{{x}_{2}} \right)}{2},\dfrac{\left( {{y}_{1}}+{{y}_{2}} \right)}{2} \right)$ and not $\left( \dfrac{\left( {{x}_{1}}-{{x}_{2}} \right)}{2},\dfrac{\left( {{y}_{1}}-{{y}_{2}} \right)}{2} \right)$ . Students often get confused between the two. Due to this confusion , they generally end up getting a wrong answer . So , such mistakes should be avoided .

Consider the figure alongside . In the question it is given that $B$ is the orthocentre. We also know that $B$ is one of the vertices of the triangle $ABC$.

So , we can conclude that $ABC$ is a right-angled triangle.

Now , we know that a right angled triangle can be inscribed in a circle with the hypotenuse as the diameter.

So , we can conclude that $AC$ is the diameter of a circle.

So , the centre of this circle will be the midpoint of the diameter.

Now, we know the midpoint of line joining the points $\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right)$ is given as:

$\left( \dfrac{\left( {{x}_{1}}+{{x}_{2}} \right)}{2},\dfrac{\left( {{y}_{1}}+{{y}_{2}} \right)}{2} \right)...............\left( i \right)$

Now , we will consider $\left( x,y \right)$ to be the coordinates of $C$.

We know , circumcentre of a triangle is the centre of the circle circumscribing the triangle.

So , the circumcentre $D\left( 2,4 \right)$ is the midpoint of the line joining $A\left( 0,0 \right)$ and $C\left( x,y \right)$.

Now , we will use the equation $\left( i \right)$ to find the coordinates of the vertex $C$.

So , from equation$\left( i \right)$ , we have

$2=\dfrac{x+0}{2}\Rightarrow x=4$

And $\text{ 4}=\dfrac{y+0}{2}\Rightarrow y=8$

So , the coordinates of vertex $C$ are $\left( 4,8 \right)$.

Option B - $\left( 4,8 \right)$ is correct answer

Note: The midpoint of line joining the points $\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right)$ is given as:

$\left( \dfrac{\left( {{x}_{1}}+{{x}_{2}} \right)}{2},\dfrac{\left( {{y}_{1}}+{{y}_{2}} \right)}{2} \right)$ and not $\left( \dfrac{\left( {{x}_{1}}-{{x}_{2}} \right)}{2},\dfrac{\left( {{y}_{1}}-{{y}_{2}} \right)}{2} \right)$ . Students often get confused between the two. Due to this confusion , they generally end up getting a wrong answer . So , such mistakes should be avoided .

Recently Updated Pages

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

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

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

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

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