Coordinate Geometry

Coordinate geometry joins algebra and geometry by describing the position of every point in a plane with an ordered pair (x, y), measured from two perpendicular axes that meet at the origin. In Class 9 you learnt to plot points in the four quadrants. In this chapter you will use coordinates to calculate distances and to divide line segments, so that geometrical facts can be checked by simple algebra instead of by measurement. The distance formula, PQ = sqrt((x2 - x1)^2 + (y2 - y1)^2), comes directly from Pythagoras theorem, and the distance of a point (x, y) from the origin is sqrt(x^2 + y^2). The section formula gives the coordinates of the point that divides the join of (x1, y1) and (x2, y2) internally in the ratio m1 : m2 as ((m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2)), with the midpoint as the special case of ratio 1 : 1. You will use these results to test collinearity, identify triangles and quadrilaterals, find points of trisection and solve map-based problems.

Detailed Worksheet: Coordinate Geometry Section A - Definitions (10 marks) 1. What are the coordinates of the origin? Write the distance of a point P(x, y) from the origin. (2 marks) 2. State the distance formula for two points A(x1, y1) and B(x2, y2). From which theorem is it derived? (2 marks) 3. State the section formula for internal division of the line segment joining (x1, y1) and (x2, y2) in the ratio m1 : m2. (2 marks) 4. Write the midpoint formula. Explain why it is a special case of the section formula. (2 marks) 5. What are points of trisection of a line segment? In what ratios do they divide the segment? (2 marks) Section B - Calculations and Applications (15 marks) 6. Find a point on the x-axis which is equidistant from the points (2, -5) and (-2, 9). (3 marks) 7. Find the values of y for which the distance between the points P(2, -3) and Q(10, y) is 10 units. (3 marks) 8. Find the coordinates of the points of trisection of the line segment joining (2, -2) and (-7, 4). (3 marks) 9. Find the ratio in which the line segment joining the points (-6, 10) and (3, -8) is divided by the point (-4, 6). (3 marks) 10. Find a relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (-3, 4). (3 marks) Section C - Diagrams (10 marks) 11. Plot the points A(-1, 2), B(1, 0), C(-1, -2) and D(-3, 0) on a graph. Join them in order and, using the distance formula, show that ABCD is a square. (4 marks) 12. Draw the line segment joining A(4, -3) and B(8, 5) on a graph and mark the point P that divides it in the ratio 3 : 1 internally. Find the coordinates of P and verify that P lies on the graph of the segment. (3 marks) 13. Plot the points (1, 2), (4, 3), (6, 6) and (3, 5) and join them in order. Using the midpoints of the diagonals, show that the figure is a parallelogram. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. In a classroom, 4 friends are seated at the points A(3, 4), B(6, 7), C(9, 4) and D(6, 1). Champa and Chameli walk into the class and, after observing for a few minutes, Champa asks Chameli, "Don't you think ABCD is a square?" Chameli disagrees. Using the distance formula, find which of them is correct. (5 marks) 15. Check whether the points (1, 5), (2, 3) and (-2, -11) are collinear, using the distance formula. Then explain, with reasons, how the distance formula can also be used to decide whether three given points form a right triangle. (5 marks) 16. The points A(1, 2), B(4, y), C(x, 6) and D(3, 5) are the vertices of a parallelogram taken in order. Find x and y. A student instead equated AB and CD using the distance formula. Explain why this method gives a harder problem and may not give unique values. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the formula used and every step of substitution. Draw all graphs on squared paper with a suitable scale.