In what ratio does the origin divides the line segment joining the points p(-1,-2,-3) and q(4,8,12)

Solution:

The coordinates of the point P(x, y) which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂), internally, in the ratio m₁: m₂ is given by the Section Formula: P(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

Let the ratio in which the line segment joining A(- 3, 10) and B(6, - 8) be divided by point C(- 1, 6) be k : 1.

By Section formula, C(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

m = k, n = 1

Therefore,

- 1 = (6k - 3) / (k + 1)

- k - 1 = 6k - 3

7k = 2

k = 2 / 7

Hence, the point C divides line segment AB in the ratio 2 : 7.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 7

Video Solution:

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 4

Summary:

The ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6) is 2 : 7.

☛ Related Questions:

Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis. Also, find the coordinates of the point of division.
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, - 3) and B is (1, 4).
If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

Math worksheets and
visual curriculum

Geo means Earth and metry means measurement. Geometry is a branch of mathematics that deals with distance, shapes, sizes, relevant positions of a figure in space, angles, and other aspects of a figure. Geometry can further be 2D or 3D geometry. 2D geometry deals with two-dimensional figures such as planes, lines, points, squares, polygons, etc. while 3D geometry is mainly concerned with three-dimensional figures or solid figures such as cubes, spheres, cuboids, etc. Let’s understand what a line segment is,

Line Segment

Line Segment is characterized by two points in space. A line segment is a line joining two distinct points in space. The distance between these 2 points is known as the length of the line segment that connects these 2 points.

Properties of a line segment

A line segment has a definite length,
A line segment cannot be extended in any direction.
A line segment has infinite points lying on it.
A line segment has a fixed name say AB, QR, etc. This name denotes the 2 points that are the end points of the line segment.

In geometry, we generally encounter problems where we are supposed to find the ratio in which a point divides a line segment. In this article, we shall discuss this concept along with an example. Consider the line segment AB as shown in the following figure. Point O lies on the line segment and divides it into 2 parts.

Let the coordinates of points A, B, and O be (x1, y1), (x2, y2) and (x, y) respectively. We are supposed to find the ratio in which the Point O divides line segment AB. Let the ratio be k : 1. To calculate the ratio we proceed using the section formula which is as follows,

and

Here, m1 = k, m2 = 1. Using these values in the above formula we get,

⇢ (1)

⇢ (2)

Use any of the equations (1) or (2) to calculate the required ratio i.e. k. Let us look at an example.

Sample Problems

Question 1: Calculate the ratio in which line segment joining the points P(1, 8) and Q(4, 2) is divided by O(3, 4).

Solution:

The line segment joining P and Q is as shown below:
Given x1 = 1, y1 = 8, x2 = 4, y2 = 2, x = 3, y = 4. Let the ratio be k : 1.
Using section formula,
Putting the values in this formula we get:
Solving for
3(k+1) = 4k+1
3k+3 = 4k+1
k = 2
Therefore, the point O divides the line segment PQ in ratio 2 : 1.

Question 2: Find the ratio in which the line segment joining the points A (-3, 3) and B (-2, 7) is divided by point O(1.5, 0).

Solution:

Given x1 = 3, y1 = 3, x2 = -2, y2 = 7, x = 1.5, y = 0
Let the required ratio be k : 1
Using Section Formula,

Solving for k, we get
1.5k+1.5 = -2k+3
3.5k = 2.5
k = 25/35 = 5/7
Therefore, the required ratio is 5:7

Question 3: Find the ratio in which the line segment joining A(1,−5) and B(−4,5) is divided by the x-axis.

Solution:

The point on the x-axis will be of the form (x, 0). Let the ratio be k:1
Given x1 = 1, y1 = -5, x2 = -4, y2 = 5, x = ?, y = 0
Using Section Formula,
0 = 5k-5
k = 1
Therefore, the required ratio is 1:1

Question 4: If a point O(x, y) divides a line segment joining A(1, 5) and B(4, 8) in two equal parts. Find point O.

Solution:

Given x1 = 1, y1 = 5, x2 = 4, y2 = 8
We are given that the line is divided into 2 equal parts by the point O. Thus the ratio is 1:1.
Now using section formula,

x= (4+1)/(1+1) = 5/2 = 2.5
Also,
y = (8+5)/(1+1) = 13/2
Therefore, the coordinates of the point O are (2.5, 6.5)