MEDIUM

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Show that the median of a triangle divides it into two triangles of equal areas

Important Questions on Quadrilaterals

HARD

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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

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MEDIUM

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

Find the centroid of the triangle, whose vertices are

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

MEDIUM

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

Find the centroid of the triangle

△ P Q R

whose vertices are

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

and

R (- 3, - 3)

MEDIUM

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

Two vertices of a triangle are

(- 1, 4)

and

(5, 2)

. If the centroid is

(0, - 3)

find the third vertex

MEDIUM

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

∆ ABC

the medians

AD, BE

and

CF

meet at

O

. What is the ratio of the area of

∆ ABD

to the area of

∆ AOE

MEDIUM

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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

(7, 1), (1, 5)

and

(1, 6)

EASY

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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

EASY

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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>Quadrilaterals>Mid Point Theorem

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>Quadrilaterals>Mid Point Theorem

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>Quadrilaterals>Mid Point Theorem

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>Quadrilaterals>Mid Point Theorem

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>Quadrilaterals>Mid Point Theorem

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:

HARD

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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 $\frac{B C}{A C}$

MEDIUM

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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>Quadrilaterals>Mid Point Theorem

The equations of the sides

A B, B C & C A

of a triangle

A B C

are

2 x + y = 0

x + p y = 21 a

(a \neq 0)

and

x - y = 3

respectively. Let

P (2, a)

be the centroid of the triangle

A B C

, then

{(B C)}^{2}

is equal to

EASY

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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 _____.

HARD

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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:

HARD

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

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}),

HARD

Mathematics>Geometry>Quadrilaterals>Mid Point Theorem

The centroid of an equilateral triangle is

(0, 0)

. If two vertices of the triangle lie on

x + y = 2 \sqrt{2}

, then one of them will have its coordinates as

Show that the median of a triangle divides it into two triangles of equal areas

Important Questions on Quadrilaterals

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