###### Advertisements

###### Advertisements

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

###### Advertisements

#### Solution 1

From the figure, it can be observed that points P, Q, R are dividing the line segment in a ratio 1:3, 1:1, 3:1 respectively.

Coordinates of P =

`((1xx2+3xx(-2))/(1+3),(1xx8+3xx2)/(1+3))`

`= (-1, 7/2)`

Coordinates of Q = `((2+(-2))/2, (2+8)/2)`

= (0,5)

Coordinates of R = `((3xx2+1xx(-2))/(3+1), (3xx8+1xx2)/(3+1))`

`=(1,13/2)`

#### Solution 2

The coordinates of the midpoint `(x_m,y_m)` between two points `(x_1,y_1)` and `(x_2,y_2)` is given by,

`(x_m,y_m) = ((x_1 + x_2)/2)"," ((y_1 + y_2)/2)`

Here we are supposed to find the points which divide the line joining A(-2,2) and B(2,8) into 4 equal parts.

We shall first find the midpoint M(x, y) of these two points since this point will divide the line into two equal parts.

`(x_m, y_m) = ((-2+2)/2)","((2+ 8)/2)`

`(x_m, y_m) = (0,5)`

So the point M(0,5) splits this line into two equal parts.

Now, we need to find the midpoint of A(-2,2) and M(0,5) separately and the midpoint of B(2,8) and M(0,5). These two points along with M(0,5) split the line joining the original two points into four equal parts.

Let M_1(e,d) be the midpoint of A(−2,2) and M(0,5).

`(e,d) = ((-2 + 0)/2)"," ((2 +5)/2)`

`(e,d) = (-1, 7/2)`

Now let `M_2(g,h)` bet the midpoint of B(2,8) and M(0,5).

`(g,h) = ((2 +0)/2)","((8 + 5)/2)`

`(g,h) = (1, 13/2)`

Hence the co-ordinates of the points which divide the line joining the two given points are (-1, 7/2), (0, 5) and (1, 13/2)

#### APPEARS IN

#### RELATED QUESTIONS

Prove that the points (–2, –1), (1, 0), (4, 3) and (1, 2) are the vertices of a parallelogram. Is it a rectangle ?

To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs `1/4` th the distance AD on the 2^{nd }line and posts a green flag. Preet runs `1/5` th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

Points A, B, C and D divide the line segment joining the point (5, -10) and the origin in five equal parts. Find the co-ordinates of B and D.

The point P (5, -4) divides the line segment AB, as shown in the figure, in the ratio 2 : 5. Find the co-ordinates of points A and B.

Show that the line segment joining the points (-5, 8) and (10, -4) is trisected by the co-ordinate axes.

A (2, 5), B (-1, 2) and C (5, 8) are the co-ordinates of the vertices of the triangle ABC. Points P

and Q lie on AB and AC respectively,

Such that: AP : PB = AQ : QC = 1 : 2

(i) Calculate the co-ordinates of P and Q.

(ii) Show that PQ =1/3BC

The line segment joining A (4, 7) and B (-6, -2) is intercepted by the y – axis at the point K. write down the abscissa of the point K. hence, find the ratio in which K divides AB. Also, find the co-ordinates of the point K.

In the given figure line APB meets the x-axis at point A and y-axis at point B. P is the point (-4,2) and AP : PB = 1 : 2. Find the co-ordinates of A and B.

In what ratio is the line joining A(0, 3) and B (4, -1) divided by the x-axis? Write the co-ordinates of the point where AB intersects the x-axis

The mid-point of the segment AB, as shown in diagram, is C(4, -3). Write down the co-ordinates of A and B.

The three vertices of a parallelogram ABCD are A(3, −4), B(−1, −3) and C(−6, 2). Find the coordinates of vertex D and find the area of ABCD.

If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (−2, 5), then the coordinates of the other end of the diameter are:

In Figure 2, P (5, −3) and Q (3, *y*) are the points of trisection of the line segment joining A (7, −2) and B (1, −5). Then *y* equals

Find the length of the hypotenuse of a square whose side is 16 cm.

The line joining P (-5, 6) and Q (3, 2) intersects the y-axis at R. PM and QN are perpendiculars from P and Q on the x-axis. Find the ratio PR: RQ.

The points A, B and C divides the line segment MN in four equal parts. The coordinates of Mand N are (-1, 10) and (7, -2) respectively. Find the coordinates of A, B and C.

In what ratio is the line joining (2, -1) and (-5, 6) divided by the y axis ?

Using section formula, show that the points A(7, −5), B(9, −3) and C(13, 1) are collinear

If point P(1, 1) divide segment joining point A and point B(–1, –1) in the ratio 5 : 2, then the coordinates of A are ______

If point P divides segment AB in the ratio 1 : 3 where A(– 5, 3) and B(3, – 5), then the coordinates of P are ______

The point Q divides segment joining A(3, 5) and B(7, 9) in the ratio 2 : 3. Find the X-coordinate of Q

Point P(– 4, 6) divides point A(– 6, 10) and B(m, n) in the ratio 2:1, then find the coordinates of point B

Find the ratio in which Y-axis divides the point A(3, 5) and point B(– 6, 7). Find the coordinates of the point

The vertices of a parallelogram in order are A(1, 2), B(4, y), C(x, 6) and D(3, 5). Then (x, y) is ______.

The points A(x_{1}, y_{1})_{,} B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are the vertices of ∆ABC. Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1

If the points A(1, –2), B(2, 3) C(a, 2) and D(– 4, –3) form a parallelogram, find the value of a and height of the parallelogram taking AB as base.

Complete the following activity to find the coordinates of point P which divides seg AB in the ratio 3:1 where A(4, – 3) and B(8, 5).

Activity:

∴ By section formula,

∴ x = `("m"x_2 + "n"x_1)/square`,

∴ x = `(3 xx 8 + 1 xx 4)/(3 + 1)`,

= `(square + 4)/4`,

∴ x = `square`,

∴ y = `square/("m" + "n")`

∴ y = `(3 xx 5 + 1 xx (-3))/(3 + 1)`

= `(square - 3)/4`

∴ y = `square`

Point C divides the line segment whose points are A(4, –6) and B(5, 9) in the ratio 2:1. Find the coordinates of C.

In what ratio does the Y-axis divide the line segment P(– 3, 1) and Q(6, 2)?