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Chapter 7: Coordinate Geometry

    Introduction

    Distance Between Two Points

    Distance Between Two Points on a Line Parallel To The Coordinate Axis

    Distance Between Any Two Points In the X-Y Plane

    Section Formula

    TriSectional Points of a Line

    Centroid Triangle

    Area of the Triangle

    Straight Lines

    Summary

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Chapter 7: Coordinate Geometry > Hard Level Worksheet

Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the coordinates of the origin. Origin = (, )

Perfect! The origin is the point where both x and y coordinates are zero.

(2) What is the distance between a point (a, b) and itself? Distance =

Correct! The distance from any point to itself is always zero.

(3) What is the shape of a figure formed by joining points with equal distances?

Excellent! All points equidistant from a center form a circle.

(4) If two points lie on the x-axis, what will be the y-coordinate of their midpoint? y-coordinate =

Perfect! Points on x-axis have y = 0, so their midpoint also has y = 0.

Short Answer Questions (2 Marks Each)

Note: Answer each question with steps and explanation, in 2-3 sentences. Write down the answers on sheet and submit to the school subject teacher.

(1) Find the distance between the points A(-3, 4) and B(5, -2). d =

Excellent application of the distance formula!

(2) Find the coordinates of the point which divides the line joining (-6, 8) and (3, -4) internally in the ratio 2:1. The point is (, )

Perfect! The dividing point is the origin.

(3) If the midpoint of the segment joining A(x, 2) and B(6, -4) is (4, -1), find the value of x. x =

Excellent! We got the answer using the midpoint formula!

(4) Show that the triangle with vertices A(1, 1), B(4, 4), C(1, 4) is a right-angled triangle.

The triangle is right-angled at .

Perfect application of Pythagoras theorem to verify right angle!

Long Answer Questions (4 Marks Each)

Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.

(1) Prove that the points A(2, 3), B(6, 7), C(10, 11) lie on the same line.

(2) Find the area of triangle formed by the points A(2, 3), B(4, 8), C(6, 3). Area =

Perfect! The area of the triangle is 10 square units.

(3) Prove that the quadrilateral with vertices A(1, 2), B(3, 4), C(6, 1), D(4, -1) is a parallelogram.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The distance between the points (–3, –4) and (3, 4) is

(a) 6 (b) 8 (c) 10 (d) 12

6
8
10
12

Correct! Distance = 3−−32+4−−42 = 36+64 = 100 = 10.

(2) If the point (x, 2x) lies on the line joining (2, 4) and (4, 8), then the value of x is

(a) 1 (b) 2 (c) 3 (d) 4

1
2
3
4

Correct! The line equation is y = 2x. Point (x, 2x) satisfies this, and checking: when x = 2, we get (2, 4) which is on the line.

(3) The point which divides the line joining (–2, –2) and (4, 4) in the ratio 1:2 is

(a) (0, 0) (b) (2, 2) (c) (3, 3) (d) (1.5, 1.5)

(0, 0)
(2, 2)
(3, 3)
(1.5, 1.5)

Correct! Using section formula: x = 1×4+2×−21+2 = 0, y = 1×4+2×−21+2 = 0.

(4) The area of triangle with vertices A(0, 0), B(4, 0), C(0, 3) is

(a) 6 (b) 12 (c) 8 (d) 4

6
12
8
4

Correct! Area = 12 |0(0-3) + 4(3-0) + 0(0-0)| = 12 |0 + 12 + 0| = 6.

(5) The centroid of a triangle with vertices (2, 3), (4, 7), (6, 1) is

(a) (4, 3.5) (b) (3, 4) (c) (4, 5) (d) (4, 113)

(4, 3.5)
(3, 4)
(4, 5)
(4, 11/3)

Centroid = (2+4+63, 3+7+13) = (123, 113) = (4, 113)

(6) Which of the following sets of points form a right-angled triangle?

(a) (0, 0), (3, 4), (3, 0) (b) (1, 1), (4, 5), (5, 6) (c) (–2, 3), (1, –1), (4, 7) (d) (2, 2), (4, 6), (6, 2)

(0, 0), (3, 4), (3, 0)
(1, 1), (4, 5), (5, 6)
(–2, 3), (1, –1), (4, 7)
(2, 2), (4, 6), (6, 2)

Correct! Sides are 5, 3, 4 and 52 = 32 + 42 (Pythagoras theorem satisfied).

(7) The length of the median drawn from the vertex of triangle A(2, 3), B(4, 5), C(6, 3) to side BC is

(a) 22 (b) 32 (c) 10 (d) None of these

2√2
3√2
√10
None of these

Correct! Midpoint of BC = (5, 4). Distance from A(2,3) to (5,4) = 5−22+4−32 = 9+1 = 10.

(8) Which of the following coordinates do not lie in the first quadrant?

(a) (3, 2) (b) (5, 0) (c) (–1, 3) (d) (1, 5)

(3, 2)
(5, 0)
(–1, 3)
(1, 5)

Correct! First quadrant has both x > 0 and y > 0. Point (-1, 3) has x < 0.

(9) The centroid of a triangle divides the median in the ratio

(a) 1:1 (b) 2:1 (c) 1:2 (d) 3:1

1:1
2:1
1:2
3:1

Correct! The centroid divides each median in the ratio 2:1 from vertex to midpoint.

(10) If a point P divides the line joining A(2, –3) and B(6, 5) in the ratio 3:1, then coordinates of P are

(a) (3, 0) (b) (5, 3) (c) (4.5, 2.5) (d) (3, 1)

(3, 0)
(5, 3)
(4.5, 2.5)
(3, 1)

Correct! Using section formula: x = 3×6+1×23+1 = 204 = 5, y = 3×5+1×−33+1 = 124 = 3.
(0,0) (2, √12) (4,0)
(1,1) (5, 1) (3,4)
(0,0) (4, 3) (2,7)
(0,0) (1, 0) (0,5)
(0,0) (3, 0) (0,4)
(2,2) (4, 2 + √12) (6,2)
Equilateral Triangle
Right Isosceles Triangle
Scalene or Irregular Triangle

Coordinate Geometry Challenge

Determine whether these statements about coordinate geometry are True or False:

Distance from a point to itself is zero
Centroid divides median in ratio 2:1
All triangles have the same area formula
Points with equal x-coordinates are collinear
Right triangle satisfies Pythagoras theorem
First quadrant contains negative coordinates

Coordinate Geometry Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Distance Formula: Calculating distances between any two points in a plane

(2) Midpoint and Section Formula: Finding points that divide line segments

(3) Area Calculations: Using coordinate formula for triangle areas

(4) Collinearity Tests: Using slopes and area methods to check if points lie on a line

(5) Shape Verification: Proving triangles are right-angled and quadrilaterals are parallelograms

(6) Geometric Properties: Understanding centroids, medians, and their relationships

(7) Quadrant Analysis: Identifying coordinate locations and their properties

(8) Problem-solving Strategies: Multiple approaches to verify geometric relationships

Excellent work mastering coordinate geometry concepts and their applications!