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