Chapter 7: Coordinate Geometry
Understanding the Coordinate Plane and Distance Formula
This chapter begins by reinforcing the basics of the coordinate plane—how to locate a point using its x- and y-coordinates. It introduces the distance formula to calculate the straight-line distance between two points in a plane. Derived using the Pythagorean theorem, this formula applies to all point pairs, whether they lie on axes or in different quadrants. The formula for the distance between two points P(x1,y1) and Q(x2,y2) is:
PQ²=(x2−x1)²+(y2−y1)²
This is used to determine whether points form a triangle, a line, or a square based on their relative distances.
Section and Midpoint Formulas
The section formula is used to find the coordinates of a point that divides a line segment joining two points in a given ratio. For a point dividing the segment internally in the ratio m1:m2, the formula is:
P(x,y) = ( m1x2+m2x1m1+m2, m1y2+m2y1m1+m2 )
It also covers how to use this formula to calculate midpoints (which divide a segment in a 1:1 ratio), find points of trisection, and solve geometry problems like checking if a quadrilateral is a square or parallelogram.
Applications and Problem Solving
The chapter applies coordinate geometry to solve various real-world problems such as locating flag positions on a sports field, dividing segments into equal parts, and identifying the intersection of a line with the x- or y-axis. Through worked examples and exercises, students gain fluency in using formulas and geometric reasoning to analyze points, lines, and shapes on the coordinate plane. The concept of internal division is emphasized, while external division is left for higher studies.