In this section we shall look at 2-Dimensional geometry in a coordinate plane. We will use a graph to examine the relationship between geometry and equations. The graph will contain two number lines – the horizontal x-axis and the vertical y-axis as seen here.
A point in the graph (specifically called the Cartesian Coordinate System) is represented with respect to its distance from the x and y axis. Every point has a unique pair of number coordinates written as (a,b), where a is the distance from the y-axis and b is the distance from the x-axis.
Over here, a is also called the x-coordinate or abscissa and b is called the y-coordinate or ordinate. In the graph below are 4 points from the 4 quadrants. Note that the perpendiculars from the points to the axes determine their abscissa and ordinate.
3.1 Distance between points
Distance between two points (x1,y1) and (x2,y2) is (x1−x2)2+(y1−y2)2.
(Note that distance is always a positive value, therefore the negative result of the square root can be ignored.)
Example 8
What is the distance between the points (−2,5) and (−7,−7)?
Solution
Distance = (−2−(−7))2+(5−(−7))2=52+122=169=13
Answer: 13
3.2 Internal and External Division of a Segment
Internal Division: Where the line segment joining two points A and B with coordinates (x1,y1) and (x2,y2) needs to be divided internally by a point P (x3,y3) such that AP : PB is in the ratio m:n, then the coordinates of P,
i.e., (x3,y3)=(m+nmx2+nx1,m+nmy2+ny1)
External Division: Where the line segment joining two points A and B with coordinates (x1,y1) and (x2,y2) need to be divided externally by a point P (x3,y3) such that AP : PB is in the ratio m:n (where m>n), then the coordinates of P,
i.e., (x3,y3)=(m−nmx2−nx1,m−nmy2−ny1)
Note that the point should lie beyond B, so that m>n. If m<n, then B will lie before A as BP > AP. In either case, the formula remains the same.
Midpoint of a Segment: The midpoint divides a line segment in the ratio of 1 : 1. Therefore, we need to substitute m=1 and n=1 in the above formula.
Midpoint (x3,y3)=(2x1+x2,2y1+y2)
Example 9
Points A, B and P are collinear. The coordinates of A are (2, 5) and B are (-3, 7). Find the coordinates of point P if P is the
(i) is the Midpoint      (ii) lies on AB and AP : PB = 2 : 5
(iii) does not lie on AB and AP : PB = 2 : 5
Solution
(i) P is the Midpoint
Let A (2,5) be (x1,y1), B (−3,7) be (x2,y2) and P be (x3,y3)
Midpoint P (x3,y3)=(2x1+x2,2y1+y2)
⇒ P(x3,y3)=(22−3,25+7)
⇒ P(x3,y3)=(2−1,6)
(ii) P lies on AB and AP : PB = 2:5
Since P (x3,y3) divides A (x1,y1) and B (x2,y2) in the ratio m:n, p(x3,y3)=(m+nmx2+nx1,m+nmy2+ny1)
Let AP : PB be m:n=2:5
⇒ P(x3,y3)=(2+52(−3)+5(2),2+52(7)+5(5))
⇒ P(x3,y3)=(7−6+10,714+25)
⇒ P(x3,y3)=(74,739)
(iii) P does not lie on AB and AP : PB = 2:5
Since P (x3,y3) divides A (x1,y1) and B (x2,y2) externally in the ratio m:n, where m>nm:n=5:2
⇒ P(x3,y3)=(m−nmx2−nx1,m−nmy2−ny1)
⇒ P(x3,y3)=(5−25(−3)−2(2),5−25(7)−2(5))
⇒ P(x3,y3)=(3−15−4,335−10)
⇒ P(x3,y3)=(3−19,325)
Answer: (i) (2−1,6), (ii) (74,739) , (iii) (3−19,325)
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