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If the vertices $P, Q, R$ of a triangle $P Q R$ , are rational points, which of the following is true?

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Important Questions on Special Points of a Triangle

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let

D

be the centroid of the triangle with vertices

(3, - 1)

(1, 3)

and

(2, 4)

. Let

P

be the point of intersection of the lines

x + 3 y - 1 = 10

and

3 x - y + 1 = 0

. Then, the line passing through the points

D

and

P

also passes through the point:

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

If the line

3 x + 4 y - 24 = 0

intersects the

x

-axis at the point

A

and the

y

-axis at the point

B,

then the incentre of the triangle

O A B,

where

O

is the origin, is:

EASY

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

R

is the circum radius of

Δ A B C

, then

A r e a (Δ A B C)

= ….

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

If a

∆ A B C

has vertices

A (- 1, 7), B (- 7, 1)

and

C (5, - 5)

, then its orthocentre has coordinates:

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let

P

be a point inside a triangle

A B C

with

∠ A B C = 90 °

. Let

P_{1} and P_{2}

be the images of

P

under reflection in

A B

and

B C

respectively. The distance between the circumcentre of triangles

A B C

and

P_{1} P P_{2}

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The incentre of the triangle with vertices

(1, \sqrt{3}), (0, 0)

and

(2, 0)

is:

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let

A (1, 0), B (6, 2)

and

C (\frac{3}{2}, 6)

be the vertices of a triangle

A B C .

P

is a point inside the triangle

A B C

such that the triangles

A P C, A P B

and

B P C

have equal areas, then the length of the line segment

P Q,

where

Q

is the point

(- \frac{7}{6}, - \frac{1}{3}),

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let a triangle

A B C

be inscribed in a circle of radius

2

units. If the

3

bisectors of the angles

A, B

and

C

are extended to cut the circle at

A_{1}, B_{1}

and

C_{1}

respectively, then the value of

{[\frac{A A_{1} \cos \frac{A}{2} + B B_{1} \cos \frac{B}{2} + C C_{1} \cos \frac{C}{2}}{\sin A + \sin B + \sin C}]}^{2} =

EASY

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points

(a^{2} + 1, a^{2} + 1)

and

(2 a, - 2 a), a \neq0

. Then for any

a,

the orthocentre of this triangle lies on the line

EASY

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let the orthocentre and centroid of a triangle be

A (- 3, 5)

and

B (3, 3)

respectively. If

C

is the circumcentre of this triangle, then the radius of the circle having line segment

A C

as diameter, is:

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The distance (in units) between the circumcentre and the centroid of the triangle formed by the vertices $(1, 2), (3, - 1)$ and $(4, 0),$ is

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let

k

be an integer such that the triangle with vertices

(k, - 3 k), (5, k)

and

(- k, 2)

has area

28

sq. units. Then the orthocenter of this triangle is at the point:

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The incentre of the triangle formed by the straight line having

3

X

-intercept and

4

Y

-intercept, together with the coordinate axes, is

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The

x -

coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as

(0, 1), (1, 1)

and

(1, 0)

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

a, b, c

are lengths of the sides

B C, C A

and

A B

respectively of

Δ A B C

and

H

is any point in the plane of

∆ A B C

such that

a \vec{A H} + b \vec{B H} + c \vec{C H} = \vec{0}

, then

H

is the

EASY

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

P (0, 0), Q (1, 0)

and

R (\frac{1}{2}, \frac{\sqrt{3}}{2})

are three given points, then the centre of the circle for which the lines

P Q, Q R

and

R P

are the tangents is

MEDIUM

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The circumcentre of the triangle with vertices at

(- 2, 3), (1, - 2)

and

(2, 1)

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let

O

be the origin and let

P Q R

be an arbitrary triangle. The point

S

is such that

\vec{O P} . \vec{O Q} + \vec{O R} . \vec{O S} = \vec{O R} . \vec{O P} + \vec{O Q} . \vec{O S} = \vec{O Q} . \vec{O R} + \vec{O P} . \vec{O S}

then triangle

P Q R

has

S

as its

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

The angle bisectors

B D

and

C E

of a

Δ A B C

are divided by the incentre

I

in the ratios

3 : 2

and

2 : 1

respectively. Then, the ratio in which

I

divides the angle bisector through

A

HARD

Mathematics>Coordinate Geometry>Special Points of a Triangle>Special Points of a Triangle

Let the equations of two sides of a triangle be

3 x - 2 y + 6 = 0

and

4 x + 5 y - 20 = 0 .

If the orthocenter of this triangle is at

(1, 1)

then the equation of it's third side is:

If the vertices P, Q, R of a triangle PQR , are rational points, which of the following is true?

Important Questions on Special Points of a Triangle

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If the vertices $P, Q, R$ of a triangle $P Q R$ , are rational points, which of the following is true?