Straight line can be defined as a simple geometrical figure. even though it is simple it is one of the important concept in geometryconcepts . We come across the concept of straight lines in our day to day experience. Sides of all polygons are straight lines. The path of the shortest distance between two points is a straight line.
Graph of a Straight Line
The graph of a linear function is a straight line. In fact the name ‘linear’ is used in the name because of this fact. The function of a straight line is expressed in several forms depending upon the context. The orientation of a straight line in a grid depicts the nature of a linear function.
A horizontal line represents a function but not a vertical line.
Distance along a Straight Line
Ratio in a Straight Line Segment
As a special case of ratio, the coordinates of the mid point of a line segment with end points as (x1, y1) and (x2, y2) are given by,
x = ½( x1 + x2) and y = ½( y1 + y2)
Area of a Triangle Formed by three Segments of Straight Lines
The area of the triangle ABC
A = Area of trapezoid DABE + Area of trapezoid EBCF - Area of trapezoid DACF
= ½ (y2 + y1)(x2 – x1) + ½ (y3 + y2)(x3 – x2) - ½ (y3+ y1)(x3 – x1)
On simplification, A = ½ [ x1 (y3 – y2) + x2 (y1 – y3) + x3(y2 – y1)]
What happens when the points A, B and C are in the same line (collinear) ?
In such a case A = 0 or, x1 (y3 – y2) + x2 (y1 – y3) + x3(y2 – y1) = 0
This brings an important concept that three points with coordinates (x1, y1), (x2, y2) and (x3, y3) will be collinear if,
x1 (y3 – y2) + x2 (y1 – y3) + x3(y2 – y1) = 0
Example Problems on Straight Lines
Example 1
The coordinates of points A and B are (-1, -3) and (2, 1) respectively. What is the distance between A and B?
Solution
As per the distance formula,
AB2 = [(2) –(-1)]2 + [(1) – (-3)]2 = (32 + 42) = (9 + 16) = 25
Hence AB = 5
The distance between the points A and B is 5 units.
Example 2
The coordinates of points P and Q are (4, 3) and (6, 7) respectively. If R is the midpoint of the straight line joining P and Q, what are the coordinates of R?
Solution
The coordinates of the mid point of a line segment with end points as (x1, y1) and (x2, y2)
are given by, x = ½( x1 + x2) and y = ½( y1 + y2)
x1 = 4, x2 = 6, y1 = 3 and y2 = 7
Therefore the coordinate of the mid point R is,
x = ½( x1 + x2) = ½( 4 + 6) = 5 and y = ½( y1 + y2) = = ½( 3 + 7) = 5
Additional Problems on Straight Lines
Problem 1
The coordinates of points P, Q and R are (2, -1), (4, 3) and (6, 7) respectively. Show that the points P, Q and R are collinear.
Solution
Three points with coordinates (x1, y1), (x2, y2) and (x3, y3) will be collinear if,
x1 (y3 – y2) + x2 (y1 – y3) + x3(y2 – y1) = 0
x1 = 2, x2 = 4, x3 = 6, y1 = -1, y2 = 3 and y3 = 7
x1 (y3 – y2) + x2 (y1 – y3) + x3(y2 – y1) = 2(7 – 3) + 4 (-1 – 7) + 6[(3 – (-1)]
= 2(4) + 4(-8) + 6(4)
= 8 – 32 + 24 = 0
Hence the points P, Q and R collinear.
Problem 2
Prove that the line joining any point on y-axis to any point on x-axis always forms a right triangle.
Solution
Assume that O is the origin, A is any point on y-axis and B be any point on x-axis.
The coordinates of A will be (0, y) and of B will be (x, 0).
Using distance formula,
OA2 = (0 – 0)2 + (y – 0)2 = y2
OB2 = (x – 0)2 + (0 – 0)2 = x2
AB2 = (0 – x)2 + (y – 0)2 = x2 + y2
Therefore, AB2 = OA2 + OB2
Hence the triangle AOB is a right triangle.
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