Coordinate Geometry
This chapter covers finding the distance between two points, the coordinates of a point dividing a line segment in a given ratio (section formula), and the area of a triangle formed by three given points.
The distance between two points A(x1, y1) and B(x2, y2) is given by: AB = root[(x2 - x1)^2 + (y2 - y1)^2]. This is derived directly from the Pythagoras theorem.
The coordinates of a point P that divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m:n are: P = [(mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)]. For the midpoint (ratio 1:1), this simplifies to P = [(x1+x2)/2, (y1+y2)/2].
Q. Find the distance between the points (3, 4) and (7, 1).
Using the distance formula: d = root[(7-3)^2 + (1-4)^2] = root[16 + 9] = root(25) = 5 units.
Area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is: (1/2) |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|. If the area comes out to 0, the three points are collinear.
Q. Find the midpoint of the line segment joining the points (2, -3) and (-6, 7).
Midpoint = [(2 + (-6))/2, (-3 + 7)/2] = [(-4)/2, (4)/2] = (-2, 2).
The distance of the point (3, 4) from the origin is:
Correct answer: (C) 5 units
If three points are collinear, the area of the triangle formed by them is _______.
Zero (0)
The distance formula can be used to prove that a given set of points forms a particular type of quadrilateral, like a square or rectangle.
True — by calculating the lengths of all sides and diagonals using the distance formula, one can verify properties like equal sides, equal diagonals, or right angles to identify the type of quadrilateral.
Find the ratio in which the point (-1, 6) divides the line segment joining A(-3, 10) and B(6, -8).
Let the ratio be k:1. Using the section formula for the x-coordinate: -1 = (6k + (-3))/(k+1), so -1(k+1) = 6k - 3, giving -k - 1 = 6k - 3, so -7k = -2, meaning k = 2/7. The ratio is 2:7.