Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 8

Coordinate Geometry

What to remember

A point is written (x, y): x is the abscissa (distance along the x-axis) and y is the ordinate; the axes divide the plane into four quadrants.
Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]; section formula and mid-point formula give points on a segment; the area of a triangle gives a test for collinearity.
Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, and perpendicular lines have slopes whose product is −1.

The Cartesian plane

René Descartes introduced the system of two perpendicular number lines. The horizontal line is the x-axis and the vertical line is the y-axis. They meet at the origin O (0, 0). The plane has four quadrants.

Quadrant	Sign of x	Sign of y
I	+	+
II	−	+
III	−	−
IV	+	−

Points on the x-axis have the form (a, 0); points on the y-axis have the form (0, b). The point (a, b) is not the same as (b, a) unless a = b. The distance of (a, b) from the x-axis is |b|, and from the y-axis is |a|.

Reflections: in the x-axis, (a, b) goes to (a, −b); in the y-axis, to (−a, b); in the origin, to (−a, −b); in the line y = x, to (b, a).

Distance formula

The distance between P(x₁, y₁) and Q(x₂, y₂) follows from the Pythagoras theorem on the right triangle formed by horizontal and vertical steps:

PQ = √[(x₂ − x₁)² + (y₂ − y₁)²].

The distance of (x, y) from the origin is √(x² + y²).

Examples: (3, 4) is 5 units from the origin. The distance between (−2, 3) and (4, −5) is √(36 + 64) = 10.

Uses of the distance formula:

Equilateral triangle: all sides equal. Isosceles: two sides equal. Right triangle: the Pythagoras relation holds for the sides.
Square: four equal sides and equal diagonals. Rhombus: four equal sides but unequal diagonals. Rectangle: opposite sides equal and equal diagonals. Parallelogram: opposite sides equal.
Collinear points A, B, C: the longest of AB, BC, CA equals the sum of the other two.
Finding a point equidistant from two given points: set the two distances equal and solve.
Circumcentre of a triangle: the point equidistant from all three vertices.

Section formula and mid-point

If P divides the segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : n, then

P = ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)).

For m = n = 1 this is the mid-point ((x₁ + x₂)/2, (y₁ + y₂)/2). For external division the denominator is m − n and signs change.

Result	Formula
Mid-point	((x₁ + x₂)/2, (y₁ + y₂)/2)
Centroid of a triangle	((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
Ratio in which the x-axis divides AB	−y₁ : y₂
Ratio in which the y-axis divides AB	−x₁ : x₂
Points of trisection	divide in ratios 1 : 2 and 2 : 1

The centroid divides every median in the ratio 2 : 1, counted from the vertex. The diagonals of a parallelogram bisect each other, so a fourth vertex can be found: D = A + C − B when ABCD is a parallelogram with B opposite to D.

Examples: The point dividing (1, 2) and (7, 8) in the ratio 1 : 2 is (3, 4). The mid-point of (2, 3) and (8, 7) is (5, 5). The centre of a circle with diameter ends (−2, −3) and (6, 5) is (2, 1).

Area of a triangle and collinearity

For vertices (x₁, y₁), (x₂, y₂), (x₃, y₃):

Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

The three points are collinear exactly when this area is zero.

Example: (1, 2), (3, 4), (5, 0) give ½|1(4 − 0) + 3(0 − 2) + 5(2 − 4)| = ½ × 12 = 6 square units.

A quadrilateral's area can be found by splitting it into two triangles along a diagonal. The area of a triangle with vertices at the origin, (a, 0) and (0, b) is ½ab.

Slope and the straight line

The slope (gradient) of the line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁) = tan θ, where θ is the angle the line makes with the positive x-axis. A horizontal line has slope 0; a vertical line has undefined slope.

Form	Equation
Slope–intercept	y = mx + c
Point–slope	y − y₁ = m(x − x₁)
Two-point	(y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)
Intercept	x/a + y/b = 1
General	Ax + By + C = 0, slope = −A/B

Parallel lines: m₁ = m₂. Perpendicular lines: m₁ m₂ = −1. The line x = k is vertical; y = k is horizontal. Three points are collinear if the slopes between pairs are equal.

Example: The line 2x + 3y = 6 has slope −2/3. A line perpendicular to a line of slope 2 has slope −1/2.

Proving shapes with coordinates

To show ABCD is a parallelogram, prove that the mid-points of AC and BD coincide, or that both pairs of opposite sides are equal. To show it is a rectangle, also show the diagonals are equal. To show it is a rhombus, show all four sides are equal. For a square, show four equal sides and equal diagonals. To show a triangle is right-angled, check that the square of the longest side equals the sum of the other two squares. To find the circumcentre, let the point be (x, y) and equate its distances to two pairs of vertices; this gives two linear equations. The area of a quadrilateral with vertices taken in order equals the sum of the areas of the two triangles formed by a diagonal.

Worked examples

1. For which k is (k, 2) at 5 units from (3, −2)? (k − 3)² + 16 = 25, so k = 6 or 0.
2. The point on the x-axis equidistant from (2, 3) and (6, 5): let it be (x, 0). Then (x − 2)² + 9 = (x − 6)² + 25, so x = 6.
3. In what ratio does the x-axis divide the join of (2, −3) and (5, 6)? Ratio = 3 : 6 = 1 : 2.
4. Three vertices of a parallelogram are A(0, 0), B(4, 0), C(5, 3). Then D = (0 + 5 − 4, 0 + 3 − 0) = (1, 3).
5. Are (1, 2), (2, 3), (3, 4) collinear? Area = ½|1(3 − 4) + 2(4 − 2) + 3(2 − 3)| = ½|−1 + 4 − 3| = 0, so yes.

Classroom angle

Start from familiar grids: seats in rows and columns, a chess board, or a street map. Use the rule "first walk along, then go up" so learners keep the order (x, y). Let them plot (2, 5) and (5, 2) to see that order matters. Treat the distance formula as the Pythagoras theorem on a grid, not as a rule to memorise. Common errors: wrong signs in subtraction, forgetting the square root, swapping the ratio in the section formula, and treating the slope of a vertical line as zero.

Exam traps

The abscissa is x and the ordinate is y; do not swap them.
Points on the y-axis have x = 0; points on the x-axis have y = 0.
Distance is never negative; the formula squares the differences first.
In the section formula, the ratio m : n multiplies the opposite point's coordinates.
Equal pairwise distances give an equilateral triangle, not collinear points.
A horizontal line has slope 0; a vertical line has undefined slope.
Parallel lines have equal slopes; perpendicular lines have slope product −1.
A rhombus has equal diagonals only if it is a square.

One-liners

1. The origin is (0, 0).
2. The quadrants are numbered anticlockwise, starting from the one with both coordinates positive.
3. The distance of (x, y) from the origin is √(x² + y²).
4. The mid-point formula averages the coordinates.
5. The centroid is the average of the three vertices.
6. The centroid divides a median 2 : 1 from the vertex.
7. The x-axis divides AB in the ratio −y₁ : y₂.
8. The area of a triangle with collinear vertices is zero.
9. Slope = (y₂ − y₁)/(x₂ − x₁).
10. Slope of ax + by + c = 0 is −a/b.
11. The reflection of (a, b) in the x-axis is (a, −b).
12. The diagonals of a parallelogram bisect each other.

Practice questions

The point (−3, 4) lies in the
1. third quadrant
2. second quadrant
3. fourth quadrant
4. first quadrant
Answer
B. second quadrant
x is negative and y is positive: quadrant II.
A point with x > 0 and y < 0 lies in the
1. third quadrant
2. fourth quadrant
3. first quadrant
4. second quadrant
Answer
B. fourth quadrant
Positive abscissa and negative ordinate give quadrant IV.
The ordinate of every point on the x-axis is
1. 1
2. the abscissa
3. undefined
4. 0
Answer
D. 0
Points on the x-axis have the form (a, 0).
The abscissa of every point on the y-axis is
1. 1
2. 0
3. equal to the ordinate
4. undefined
Answer
B. 0
Points on the y-axis have the form (0, b).
The distance of the point (3, 4) from the origin is
1. 5
2. 3.5
3. 25
4. 7
Answer
A. 5
√(3² + 4²) = 5.
The distance between the points (1, 2) and (4, 6) is
1. 3
2. 7
3. √7
4. 5
Answer
D. 5
√(3² + 4²) = 5.
The distance between (−2, 3) and (4, −5) is
1. √28
2. 14
3. 10
4. 8
Answer
C. 10
√(6² + 8²) = √100 = 10.
The distance of the point (−5, 12) from the origin is
1. 13
2. 7
3. 169
4. 17
Answer
A. 13
√(25 + 144) = 13.
The distance of the point (3, −4) from the x-axis is
1. 3
2. 5
3. 4
4. −4
Answer
C. 4
The distance from the x-axis is the absolute value of the ordinate.
The mid-point of the segment joining (2, 3) and (8, 7) is
1. (10, 10)
2. (5, 5)
3. (3, 2)
4. (6, 4)
Answer
B. (5, 5)
((2 + 8)/2, (3 + 7)/2) = (5, 5).
The point dividing the line joining (1, 2) and (7, 8) internally in the ratio 1 : 2 is
1. (5, 6)
2. (4, 5)
3. (3, 4)
4. (2, 3)
Answer
C. (3, 4)
x = (1·7 + 2·1)/3 = 3; y = (1·8 + 2·2)/3 = 4.
The point that is one-third of the way from (0, 0) to (9, 12) is
1. (6, 8)
2. (3, 3)
3. (4.5, 6)
4. (3, 4)
Answer
D. (3, 4)
Divide in the ratio 1 : 2: (9/3, 12/3).
The centroid of the triangle with vertices (0, 0), (6, 0) and (0, 9) is
1. (2, 3)
2. (6, 9)
3. (2, 4.5)
4. (3, 4.5)
Answer
A. (2, 3)
((0 + 6 + 0)/3, (0 + 0 + 9)/3) = (2, 3).
The centroid divides each median in the ratio
1. 2 : 1 from the vertex
2. 1 : 1
3. 1 : 2 from the vertex
4. 3 : 1 from the vertex
Answer
A. 2 : 1 from the vertex
The centroid is two-thirds of the way from the vertex.
The area of the triangle with vertices (0, 0), (4, 0) and (0, 6) is
1. 10 square units
2. 24 square units
3. 6 square units
4. 12 square units
Answer
D. 12 square units
½ × 4 × 6 = 12.
The area of the triangle with vertices (1, 2), (3, 4) and (5, 0) is
1. 10 square units
2. 4 square units
3. 6 square units
4. 12 square units
Answer
C. 6 square units
½|1(4 − 0) + 3(0 − 2) + 5(2 − 4)| = ½|4 − 6 − 10| = 6.
The points (1, 2), (2, 3) and (3, k) are collinear. Then k is
1. 5
2. 3
3. 4
4. 6
Answer
C. 4
The area is zero when k = 4 (slope 1 throughout).
The slope of the line through (1, 2) and (3, 6) is
1. 2
2. 4
3. 1/2
4. −2
Answer
A. 2
(6 − 2)/(3 − 1) = 2.
The slope of the line 2x + 3y = 6 is
1. 2/3
2. −3/2
3. 2
4. −2/3
Answer
D. −2/3
y = −(2/3)x + 2, so the slope is −2/3.
The slope of a line perpendicular to a line of slope 2 is
1. 1/2
2. −1/2
3. −2
4. 2
Answer
B. −1/2
The product of slopes of perpendicular lines is −1.
The equation of the line with slope 3 and y-intercept 2 is
1. y = −3x + 2
2. y = 3x + 2
3. y = 2x + 3
4. y = 3x − 2
Answer
B. y = 3x + 2
y = mx + c with m = 3 and c = 2.
The equation of the line parallel to the x-axis through (3, 5) is
1. x = 5
2. y = 3
3. x = 3
4. y = 5
Answer
D. y = 5
A horizontal line has constant y.
The reflection of the point (3, −4) in the x-axis is
1. (4, 3)
2. (3, 4)
3. (−3, −4)
4. (−3, 4)
Answer
B. (3, 4)
Reflection in the x-axis changes the sign of the ordinate.
The reflection of the point (3, −4) in the origin is
1. (−3, −4)
2. (3, 4)
3. (−3, 4)
4. (−4, 3)
Answer
C. (−3, 4)
Reflection in the origin changes both signs.
The point on the x-axis equidistant from (2, 3) and (6, 5) is
1. (6, 0)
2. (0, 6)
3. (4, 0)
4. (5, 0)
Answer
A. (6, 0)
With (x, 0): (x−2)² + 9 = (x−6)² + 25 gives x = 6.
The x-axis divides the line joining (2, −3) and (5, 6) in the ratio
1. 1 : 1
2. 1 : 2
3. 2 : 1
4. 3 : 2
Answer
B. 1 : 2
Ratio = −y₁ : y₂ = 3 : 6 = 1 : 2.
The y-axis divides the line joining (−2, 5) and (4, 7) in the ratio
1. 1 : 2
2. 1 : 1
3. 2 : 1
4. 2 : 3
Answer
A. 1 : 2
Ratio = −x₁ : x₂ = 2 : 4 = 1 : 2.
The perimeter of the triangle with vertices (0, 0), (3, 0) and (0, 4) is
1. 14 units
2. 10 units
3. 7 units
4. 12 units
Answer
D. 12 units
Sides are 3, 4 and 5.
The triangle with vertices (0, 0), (4, 0) and (2, 2√3) is
1. right-angled
2. isosceles but not equilateral
3. equilateral
4. scalene
Answer
C. equilateral
All three sides equal 4.
Three vertices of a parallelogram ABCD are A(0, 0), B(4, 0) and C(5, 3). The fourth vertex D is
1. (1, −3)
2. (1, 3)
3. (9, 3)
4. (−1, 3)
Answer
B. (1, 3)
D = A + C − B = (1, 3).
The value(s) of k for which the point (k, 2) is 5 units from (3, −2) are
1. 0 or 6
2. 6 only
3. 3 only
4. −6 or 0
Answer
A. 0 or 6
(k−3)² + 16 = 25 gives k − 3 = ±3.
One end of a segment is (1, 2) and its mid-point is (3, 4). The other end is
1. (4, 5)
2. (2, 3)
3. (5, 6)
4. (7, 8)
Answer
C. (5, 6)
Other end = (2·3 − 1, 2·4 − 2).
The centre of a circle whose diameter has ends (−2, −3) and (6, 5) is
1. (0, 0)
2. (4, 8)
3. (4, 2)
4. (2, 1)
Answer
D. (2, 1)
The centre is the mid-point: (4/2, 2/2).
The radius of the circle whose diameter has ends (−2, −3) and (6, 5) is
1. 4√2 units
2. 8 units
3. 8√2 units
4. 4 units
Answer
A. 4√2 units
Diameter = √(64 + 64) = 8√2, so radius = 4√2.
Consider the statements: 1. The abscissa of a point is its x-coordinate. 2. Every point on the y-axis has abscissa zero. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard definitions.
Consider the statements: 1. The distance formula follows from the Pythagoras theorem. 2. The distance between two points can be negative. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Distance is never negative.
Consider the statements: 1. The mid-point formula is a special case of the section formula with ratio 1 : 1. 2. The centroid of a triangle is the mean of the coordinates of its vertices. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are true.
Consider the statements: 1. Three points are collinear if the area of the triangle they form is zero. 2. Three points are collinear if all pairwise distances are equal. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Equal pairwise distances give an equilateral triangle, not a line.
Consider the statements: 1. The slope of the x-axis is undefined. 2. The slope of a vertical line is undefined. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The x-axis is horizontal with slope 0.
Consider the statements: 1. The points (a, b) and (b, a) are always the same point. 2. The point (0, −3) lies on the x-axis. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
They coincide only if a = b; (0, −3) lies on the y-axis.
Which pair is correctly matched?
1. Distance : (x₂ − x₁) + (y₂ − y₁)
2. Centroid : ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
3. Slope : (x₂ − x₁)/(y₂ − y₁)
4. Mid-point : ((x₁ − x₂)/2, (y₁ − y₂)/2)
Answer
B. Centroid : ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
The others have wrong formulas.
To help Class IX learners write coordinates in the correct order, a teacher should
1. tell learners to write either order
2. use a grid or seating-plan activity and the rule: first walk along, then go up
3. avoid using graph paper
4. only use negative numbers first
Answer
B. use a grid or seating-plan activity and the rule: first walk along, then go up
A concrete grid links (x, y) to movement along then up.
A learner always plots (y, x) instead of (x, y). The most helpful step is to
1. give a lecture on quadrants
2. ask the learner to memorise the formula
3. deduct marks without feedback
4. plot (2, 5) and (5, 2) on a grid and compare the positions
Answer
D. plot (2, 5) and (5, 2) on a grid and compare the positions
Seeing that the points differ shows why order matters.
The point on the y-axis that is 3 units below the origin is
1. (0, 3)
2. (−3, 0)
3. (0, −3)
4. (3, 0)
Answer
C. (0, −3)
On the y-axis x = 0; below the origin y is negative.

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Chapter quiz · 10 questions

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1.The area of quadrilateral is 336 m 2 and the perpendiculars drawn to one diagonal from the opposite vertices are 16 m and 12 m long. Find the length of this diagonal
2.The base of a parallelogram is twice as long as its corrosponding height. If the area of the parallelogram is 144 cm², find the mentioned height
3.The given pie chart shows the percentage wise distribution of books in 5 different states A, B, C, D and E. The total number of books is 2500.
4.The Earth makes a complete rotation about its axis in 24 hours. Through what angle will it turn in 3 hours 30 minutes?
5.The ratio of measures of the angles of a triangle is given as 3: 4: 8. What is the measure of the smallest of the three angles?
6.The radii of two circles are 4 cm and 7 cm and the distance between their centres is 12 cm. Find the length of the direct common tangent to the circles.
7.The ratio in which a transverse common tangent drawn to two circles with radii 4 cm and 6 cm, respectively, divides the line joining their centres is:
8.The length of the tangent drawn to a circle of radius 12 cm from a point 18 cm away from the centre of the circle is:
9.PQ and RS are two parallel chords of a circle of length 10 cm and 24 cm, respectively, and lie on the same side of the centre O. If the distance between the chords is 7 cm, what is the radius (in cm) of the circle?
10.RA and RB are two tangents from a common point 'R' outside the circle with centre 'O', having a radius of 6 cm. If ARB = 70°, find the measure of the angle AOB.

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