HARD

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Define centroid of a triangle. Write its properties and the formula for centroid of a triangle.

Important Questions on Coordinate Geometry

EASY

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

The vertices of a triangle are

(3, - 5)

and

(- 7, 4)

. If its centroid is

(2, - 1),

then the third vertex is

HARD

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

Let

a, b, c

be in arithmetic progression. Let the centroid of the triangle with vertices

(a, c), (2, b)

and

(a, b)

(\frac{10}{3}, \frac{7}{3}) .

α, β

are the roots of the equation

a x^{2} + b x + 1 = 0,

then the value of

α^{2} + β^{2} - α β

is:

MEDIUM

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

Let

D, E, F

be points on the sides

B C, C A, A B

of a triangle

A B C,

respectively. Suppose

A D, B E, C F

are concurrent at

P .

\frac{P F}{P C} = \frac{2}{3}, \frac{P E}{P B} = \frac{2}{7}

and

\frac{P D}{P A} = \frac{m}{n}

where

m, n

are positive integers with

gcd (m, n) = 1 .

Find

(m + n)

EASY

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

Centroid of a triangle, whose vertices are $(- a, 0), (0, b) and (a, 0)$ is _____

MEDIUM

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

In a triangle

A B C

, the medians

A D

and

B E

intersect at

G

. A line

D F

is drawn parallel to

B E

such that

F

is on

A C

. If

A C = 9 cm

, then what is

C F

equal to?

MEDIUM

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

Find the centroid of the triangle, whose vertices are

(- 4, 4), (- 2, 2) and (- 6, 6)

MEDIUM

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

∆ ABC

the medians

AD, BE

and

CF

meet at

O

. What is the ratio of the area of

∆ ABD

to the area of

∆ AOE

HARD

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

In an isosceles triangle

A B C

, the vertex

A

(6, 1)

and the equation of the base

B C

2 x + y = 4

. Let the point

B

lie on the line

x + 3 y = 7

. If

(α, β)

is the centroid

∆ A B C

, then

15 (α + β)

is equal to

MEDIUM

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

Suppose

∆ A B C

is an isosceles triangle with

∠ C = 90 °, A = (2, 3)

and

B = (4, 5)

. Then the centroid of the triangle is

HARD

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

Find the centroid of the triangle

△ P Q R

whose vertices are

P (1, 1), Q (2, 2)

and

R (- 3, - 3)

HARD

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

Let $A B C D$ be a rectangle and $M$ be the midpoint of $C D$ . Suppose $A C$ and $B D$ meet at $O, A M$ and $B D$ meet at $X$ , $B M$ and $A C$ meet at $Y$ . Then

Question Image

HARD

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

Let $G$ be the centroid of the triangle $A B C$ in which the angle $C$ is obtuse. Let $A D$ and $C F$ are the medians from $A$ and $C$ on the sides $B C$ and $A B$ respectively. If the four points $B, D, G$ and $F$ are concyclic, then $\frac{B C}{A C}$

EASY

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

A (2, 4), B (3, 5), C (4, 3)

are the vertices of

△ A B C

. Hence the coordinate of centroid of the triangle is _____.

MEDIUM

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

Two vertices of a triangle are

(- 1, 4)

and

(5, 2)

. If the centroid is

(0, - 3)

find the third vertex

EASY

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

The centroid of the triangle formed by the lines

x - 3 y + 3 = 0, x + 3 y + 3 = 0, x + y - 1 = 0

MEDIUM

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

The distance of the origin from the centroid of the triangle whose two sides have the equations

x - 2 y + 1 = 0

and

2 x - y - 1 = 0

and whose orthocenter is

(\frac{7}{3}, \frac{7}{3})

is:

MEDIUM

Mathematics>Geometry>Coordinate Geometry>Special Points in 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:

MEDIUM

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

The distance from the origin to the centroid of the triangle whose vertices are

(7, 1), (1, 5)

and

(1, 6)

MEDIUM

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

$X M$ is a median in $△ X Y Z, X Y^{2} + X Z^{2} = 328$ and $X M = 8$ then $Y Z =$ _____.

Define centroid of a triangle. Write its properties and the formula for centroid of a triangle.

Important Questions on Coordinate Geometry

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