A ( 4 , 2 ) , B ( 6 , 5 ) and C ( 1 , 4 ) are the vertices of △ A B C . The median from A meets B C in D . Find the coordinates of the point D .

We have triangle △ A B C in which the co-ordinates of the vertices are A ( 4 , 2 ) ; B ( 6 , 5 ) and C ( 1 , 4 ) . AD is the median from A to B C . So, D is the mid-point of side B C . In general to find the mid-point P ( x , y ) of two points A x 1 , y 1 and B ( x 2 , y 2 ) we use formula as given below: P ( x , y ) = x 1 + x 2 2 , y 1 + y 2 2 Therefore mid-point D of side B C can be written as, D ( x , y ) = 6 + 1 2 , 5 + 4 2 Now equate the individual terms to get, x = 7 2 , y = 9 2 So co-ordinates of D is 7 2 , 9 2 .

A point P divides the line segment joining the points A ( 3 , − 5 ) and B ( − 4 , 8 ) such that A P P B = k 1 . If P lies on the line x + y = 0 , then find the value of k .

It is given that A P P B = k 1 . So, P divides the line segment joining the points A ( 3 , − 5 ) and B ( − 4 , 8 ) in the ratio k : 1 The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m : n is P ( x , y ) = n x 1 + m x 2 m + n , n y 1 + m y 2 m + n . Using the section formula, we get Coordinates of P = - 4 k + 3 k + 1 , 8 k - 5 k + 1 Since P lies on the line x + y = 0 , so - 4 k + 3 k + 1 + 8 k - 5 k + 1 = 0 ⇒ - 4 k + 3 + 8 k - 5 k + 1 = 0 ⇒ 4 k - 2 = 0 ⇒ k = 1 2 Hence, the value of k is 1 2 .

A ( 4 , 2 ) , B ( 6 , 5 ) and C ( 1 , 4 ) are the vertices of △ A B C . The median from A meets B C at D = 7 2 , 9 2 . Find the coordinates of point P on A D such that A P : P D = 2 : 1 .

Given, A ( 4 , 2 ) , B ( 6 , 5 ) and C ( 1 , 4 ) are the vertices of △ A B C . The median from A meets B C at D = 7 2 , 9 2 . We draw the figure using the given conditions as, We have to find the coordinates of a point P , which divides A D in the ratio 2 : 1 , internally. According to the section formula, if a point P divides a line segment joining A x 1 , y 1 and D x 2 , y 2 in the ratio m : n internally, then P ( x , y ) = n x 1 + m x 2 m + n , n y 1 + m y 2 m + n . P divides A D in the ratio 2 : 1 . So, P x , y = 4 + 2 7 2 1 + 2 , 2 + 2 9 2 1 + 2 = 11 3 , 11 3 Therefore, the coordinates of P are 11 3 , 11 3 .

EASY

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The theorem that we use in finding the length of space diagonal in rectangular prism is ?

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Important Questions on Space

MEDIUM

Mathematics>Geometry>Space>Describing Space

A (4, 2), B (6, 5) and C (1, 4)

are the vertices of

△ A B C .

The median from

A

meets

B C

D

. Find the coordinates of the point

D

MEDIUM

Mathematics>Geometry>Space>Describing Space

Determine the ratio in which the straight line $x - y - 2 = 0$ divides the line segment joining $(3, - 1)$ and $(8, 9)$ .

MEDIUM

Mathematics>Geometry>Space>Describing Space

Points

P, Q, R and S

divides the line segment joining

A (1, 2) and B (6, 7) in 5

equal parts. Find the coordinates of the points

P, Q and R .

MEDIUM

Mathematics>Geometry>Space>Describing Space

The line joining the points

(2, 1) and (5, - 8)

is trisected at the points

P and Q .

If point

P

lies on the line

2 x - y + k = 0 .

Find the value of

k

MEDIUM

Mathematics>Geometry>Space>Describing Space

If three consecutive vertices of a parallelogram are

(1, - 2), (3, 6) and (5, 10),

find its fourth vertex.

HARD

Mathematics>Geometry>Space>Describing Space

A B C D

is a rectangle formed by joining the points

A (- 1, - 1)

B (- 1, 4)

C (5, 4)

, and

D (5, - 1)

P, Q, R and S

are the mid-points of sides

A B, B C, C D

and

D A

respectively. Is the quadrilateral

P Q R S

a square? a rectangle? or a rhombus? Justify your answer.

MEDIUM

Mathematics>Geometry>Space>Describing Space

A point

P

divides the line segment joining the points

A (3, - 5) and B (- 4, 8)

such that

\frac{A P}{P B} = \frac{k}{1}

. If

P

lies on the line

x + y = 0,

then find the value of

k

MEDIUM

Mathematics>Geometry>Space>Describing Space

Show that the mid-point of the line segment joining the points

(5, 7)

and

(3, 9)

is also the mid-point of the line segment joining the points

(8, 6)

and

(0, 10)

HARD

Mathematics>Geometry>Space>Describing Space

If the coordinates of the mid-points of the sides of a triangle are

(3, 4), (4, 6), and (5, 7)

, then find its vertices.

MEDIUM

Mathematics>Geometry>Space>Describing Space

A

and

B

are

(1, 4)

and

(5, 2)

respectively, find the coordinates of

P

when

\frac{A P}{B P} = \frac{3}{4}

MEDIUM

Mathematics>Geometry>Space>Describing Space

If the points

A (a, - 11), B (5, b), C (2, 15) and D (1, 1)

are the vertices of a parallelogram

A B C D

, find the values of

a

and

b

MEDIUM

Mathematics>Geometry>Space>Describing Space

The line segment joining the points

(3, - 4) and (1, 2)

is trisected at the points

P

and

Q

. If the coordinates of

P

and

Q

are

(p, - 2) and (\frac{5}{3}, q)

respectively. Find the values of

p

and

q

MEDIUM

Mathematics>Geometry>Space>Describing Space

A

and

B

are two points having coordinates

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

respectively, find the coordinates of

P

such that

A P = \frac{3}{7} A B

MEDIUM

Mathematics>Geometry>Space>Describing Space

$A (4, 2), B (6, 5) and C (1, 4)$ are the vertices of $△ A B C$ . The median from $A$ meets $B C$ at $D = (\frac{7}{2}, \frac{9}{2})$ . Find the coordinates of point $P$ on $A D$ such that $A P : P D = 2 : 1$ .

MEDIUM

Mathematics>Geometry>Space>Describing Space

Determine the ratio in which the point

(- 6, a)

divides the join of

A (- 3, 1)

and

B (- 8, 9)

. Also, find the value of

a

MEDIUM

Mathematics>Geometry>Space>Describing Space

If two vertices of a parallelogram are

(3, 2) (- 1, 0)

and the diagonals cut at

(2, - 5)

, find the other vertices of the parallelogram.

EASY

Mathematics>Geometry>Space>Describing Space

Cotyledons are also called-

MEDIUM

Mathematics>Geometry>Space>Describing Space

Determine the ratio in which the point

P (m, 6)

divides the join of

A (- 4, 3)

and

B (2, 8)

. Also, find the value of

m

MEDIUM

Mathematics>Geometry>Space>Describing Space

Find the coordinates of the points which divide the line segment joining

A (- 2, 2) and B (2, 8)

into four equal parts.

HARD

Mathematics>Geometry>Space>Describing Space

If the coordinates of the mid-point of the sides of a triangle be

(3, - 2), (- 3, 1) and (4, - 3),

then find the coordinates of its vertices.

The theorem that we use in finding the length of space diagonal in rectangular prism is ?

Important Questions on Space

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