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State Board · Class 10 · Mathematics Part 2

Lesson 5: Co-ordinate Geometry

3 concepts · 24 practice questions with answers · Free to read

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Distance between two points

From daily life

A map of a school is drawn on graph paper. We want the straight distance between the classroom and the playground. If we know the horizontal and the vertical distance, Pythagoras gives the straight distance.

Simple definition

The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The distance of P(x, y) from the origin is √(x² + y²).

xy4353² + 4² = 5²

In the figure: Horizontal distance 4, vertical distance 3; straight distance = √(16 + 9) = 5.

Rule 1: Use the distance formula — Find the difference of the x values and of the y values, square both, add and take the square root.

  • Easy: A(1, 2), B(4, 6) ⇒ differences 3 and 4; d = √(9 + 16) = 5.
  • Medium: P(−2, 3), Q(4, −5) ⇒ differences 6 and −8; d = √(36 + 64) = 10.
  • Exam-type: A(0, 0), B(1, 1) ⇒ d = √(1 + 1) = √2; and C(3, 7), D(5, 3) ⇒ differences 2 and −4; d = √(4 + 16) = √20 = 2√5.

Rule 2: Distance from the origin — The origin is O(0, 0), so the distance of P(x, y) is √(x² + y²).

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Distance between two points

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  • Easy: P(6, 8) ⇒ √(36 + 64) = 10.
  • Medium: P(5, 12) ⇒ √(25 + 144) = 13.
  • Exam-type: P(9, −12) ⇒ √(81 + 144) = 15; the minus sign vanishes on squaring.

Rule 3: Are three points in a line? — Find the three lengths. If the sum of the two smaller lengths equals the biggest, the points are collinear (in one line).

  • Easy: A(0, 1), B(2, 3), C(5, 6) ⇒ AB = √8 = 2√2, BC = √18 = 3√2, AC = √50 = 5√2; 2√2 + 3√2 = 5√2, so they are collinear.
  • Medium: P(1, 1), Q(4, 1), R(1, 5) ⇒ PQ = 3, PR = 4, QR = 5; 3 + 4 ≠ 5, so they are not collinear.
  • Exam-type: P(0, 0), Q(3, 0), R(0, 4): 3² + 4² = 5², so the triangle is right-angled.
Remember

Keep the order of the points the same: x₂ − x₁ and y₂ − y₁.

Common mistake

Do not take the root before adding: √(9 + 16) ≠ 3 + 4.

Shortcut

When you see the triplets (3, 4, 5), (5, 12, 13) and (6, 8, 10), write the answer without a root. Memory clue: "difference, square, add, root".

Spot the difference

PointDistanceIn a line
What to findLength between two pointsRelation of three lengths
Formula√[(x₂ − x₁)² + (y₂ − y₁)²]small + small = big
From the origin√(x² + y²)O used as a point
Right anglea² + b² = c²small² + small² = big²

Practice questions

Easy
  1. 1What is the distance between A(0, 0) and B(3, 4)?
  2. 2What is the distance between P(2, 3) and Q(2, 9)?
  3. 3What is the distance of P(8, 15) from the origin?
Medium
  1. 4Find the distance between A(1, 1) and B(7, 9).
  2. 5Find the distance between A(−3, 1) and B(2, −11).
  3. 6Are A(2, 2), B(5, 6) and C(8, 10) collinear?
Challenging
  1. 7The point (x, 0) is at a distance 5 from A(1, 3). Find x.
  2. 8What kind of triangle is formed by A(0, 0), B(6, 0), C(3, 4)?
Show answers (try them yourself first)
  1. 1√(9 + 16) = 5.
  2. 2x is the same; d = 9 − 3 = 6.
  3. 3√(64 + 225) = 17.
  4. 4Differences 6 and 8; d = 10.
  5. 5Differences 5 and −12; d = √(25 + 144) = 13.
  6. 6AB = 5, BC = 5, AC = 10; 5 + 5 = 10, so yes.
  7. 7(x − 1)² + 9 = 25 ⇒ (x − 1)² = 16 ⇒ x = 5 or −3.
  8. 8AB = 6, AC = 5, BC = 5; two sides are equal, so it is isosceles.

Section formula

From daily life

The road between two villages is 12 km. A petrol pump is at the ratio 1 : 2 from the first village, which means at 4 km. The co-ordinates of such a dividing point are found by a formula.

Simple definition

If the point 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)). If m = n, then P is the midpoint.

121 : 2

In the figure: The segment AB divided in the ratio 1 : 2; P is nearer to A and AP is half of PB.

Rule 1: Midpoint — The average of the two x values and of the two y values: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

  • Easy: A(2, 4), B(8, 10) ⇒ ((2 + 8) ÷ 2, (4 + 10) ÷ 2) = (5, 7).
  • Medium: A(−4, 6), B(2, −2) ⇒ (−2 ÷ 2, 4 ÷ 2) = (−1, 2).
  • Exam-type: Midpoint (3, 5) and A(1, 2) ⇒ B: 2 × 3 − 1 = 5 and 2 × 5 − 2 = 8, so B(5, 8).

Rule 2: Division in the ratio m : n — For each co-ordinate: (m × the second + n × the first) ÷ (m + n).

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Section formula

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  • Easy: A(1, 2), B(7, 8), 1 : 2 ⇒ x = (1 × 7 + 2 × 1) ÷ 3 = 3; y = (1 × 8 + 2 × 2) ÷ 3 = 4; P(3, 4).
  • Medium: A(−3, 5), B(6, −1), 2 : 1 ⇒ x = (2 × 6 + 1 × (−3)) ÷ 3 = 3; y = (2 × (−1) + 1 × 5) ÷ 3 = 1; P(3, 1).
  • Exam-type: A(2, −3), B(10, 5), 3 : 1 ⇒ x = (3 × 10 + 1 × 2) ÷ 4 = 8; y = (3 × 5 + 1 × (−3)) ÷ 4 = 3; P(8, 3).

Rule 3: Centroid of a triangle — The point where the three medians meet is the centroid G. G = ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3). It divides each median in the ratio 2 : 1.

  • Easy: A(1, 2), B(4, 6), C(7, 1) ⇒ G = (12 ÷ 3, 9 ÷ 3) = (4, 3).
  • Medium: A(−2, 0), B(4, 3), C(1, 9) ⇒ G = (3 ÷ 3, 12 ÷ 3) = (1, 4).
  • Exam-type: A(0, 0), B(6, 0), C(3, 9) ⇒ G = (9 ÷ 3, 9 ÷ 3) = (3, 3).
Remember

In the midpoint the ratio is 1 : 1; for the centroid we average three points.

Common mistake

Do not swap m and n: m is multiplied with the co-ordinate of the second point.

Shortcut

A dividing point always lies between the two points; check it. Memory clue: "cross multiply" — m with the far end, n with the near end.

Spot the difference

PointMidpointCentroid
Points23
Divide by÷ 2÷ 3
Formula((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3)
Ratio1 : 12 : 1

Practice questions

Easy
  1. 1Find the midpoint of A(2, 6) and B(4, 10).
  2. 2Find the midpoint of A(−2, 0) and B(2, 4).
  3. 3Find the centroid of A(0, 0), B(6, 9), C(3, 0).
Medium
  1. 4A(1, 1), B(7, 4); P divides AB in the ratio 1 : 2. Find P.
  2. 5A(−1, 7), B(4, −3); find the point dividing in the ratio 2 : 3.
  3. 6The midpoint of AB is (4, 5) and A is (1, 2). Find B.
Challenging
  1. 7Two vertices of a triangle are (2, 1) and (6, 5) and the centroid is (4, 3). Find the third vertex.
  2. 8A(2, 3) and B(8, 9); P is the one-third point (1 : 2 from A) and Q is at 2 : 1. Find P and Q.
Show answers (try them yourself first)
  1. 1(3, 8).
  2. 2(0, 2).
  3. 3(9 ÷ 3, 9 ÷ 3) = (3, 3).
  4. 4x = (7 + 2) ÷ 3 = 3; y = (4 + 2) ÷ 3 = 2; P(3, 2).
  5. 5x = (2 × 4 + 3 × (−1)) ÷ 5 = 1; y = (2 × (−3) + 3 × 7) ÷ 5 = 3; (1, 3).
  6. 6B = (2 × 4 − 1, 2 × 5 − 2) = (7, 8).
  7. 7x = 3 × 4 − 2 − 6 = 4; y = 3 × 3 − 1 − 5 = 3; (4, 3).
  8. 8P = ((8 + 4) ÷ 3, (9 + 6) ÷ 3) = (4, 5); Q = ((16 + 2) ÷ 3, (18 + 3) ÷ 3) = (6, 7).

Slope of a line

From daily life

How steep a staircase is can be told by height ÷ width. More height means a steeper slope. The slope of a line is measured the same way.

Simple definition

The slope of the line through A(x₁, y₁) and B(x₂, y₂) is m = (y₂ − y₁) ÷ (x₂ − x₁) = rise ÷ run. If the line makes an angle θ with the X-axis then m = tan θ.

xy422 ÷ 4= 1/2

In the figure: Run 4, rise 2; slope = 2 ÷ 4 = 1/2.

Rule 1: The slope formula — Rise ÷ run. Keep the order of both differences the same.

  • Easy: A(1, 2), B(3, 6) ⇒ (6 − 2) ÷ (3 − 1) = 4 ÷ 2 = 2.
  • Medium: A(2, 5), B(6, 1) ⇒ (1 − 5) ÷ (6 − 2) = −4 ÷ 4 = −1; the line goes down.
  • Exam-type: A(−1, 3), B(5, 12) ⇒ (12 − 3) ÷ (5 + 1) = 9 ÷ 6 = 3/2.

Rule 2: Lines parallel to axes, and slope from the angle — A horizontal line has slope 0; a vertical line has no slope (undefined). If the angle θ is given, m = tan θ.

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Slope of a line

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  • Easy: (2, 5) and (7, 5) ⇒ rise 0; slope 0 ÷ 5 = 0.
  • Medium: (3, 1) and (3, 9) ⇒ run 0; 8 ÷ 0 is not possible, so the slope is undefined.
  • Exam-type: θ = 45° ⇒ m = 1; θ = 60° ⇒ m = √3; θ = 30° ⇒ m = 1/√3.

Rule 3: Parallel lines and collinear points — Parallel lines have equal slopes. If the slopes of AB and BC are equal for three points A, B, C, they lie in one line.

  • Easy: AB: A(0, 0), B(2, 3) slope 3/2; CD: C(4, 1), D(8, 7) slope 6/4 = 3/2 ⇒ the lines are parallel.
  • Medium: P(1, 2), Q(3, 6), R(5, 10) ⇒ slope of PQ is 4 ÷ 2 = 2; slope of QR is 4 ÷ 2 = 2 ⇒ collinear.
  • Exam-type: A(0, 1), B(2, 3), C(4, 4) ⇒ slope of AB is 1, slope of BC is 1/2 ⇒ not collinear.
Remember

Rise on top and run below: "rise over run".

Common mistake

Do not reverse the order of the points in only one difference; the sign of the slope goes wrong.

Shortcut

A positive slope goes up, a negative slope goes down. Memory clue: "horizontal 0, vertical undefined".

Spot the difference

PointHorizontal lineVertical line
Rise0Any
RunAny0
Slope0Undefined
Angle0°90°

Practice questions

Easy
  1. 1What is the slope of the line through A(0, 0) and B(2, 6)?
  2. 2What is the slope of the line through (1, 4) and (5, 4)?
  3. 3What is the slope of a line at an angle of 45°?
Medium
  1. 4Find the slope of A(2, 7), B(6, 3).
  2. 5Find the slope of A(−2, 1) and B(4, 10).
  3. 6Are A(1, 2), B(2, 5), C(4, 11) collinear?
Challenging
  1. 7The line through A(1, 1) and B(3, k) has slope 4. Find k.
  2. 8Are the lines through P(0, 3), Q(4, 7) and R(1, 0), S(5, 4) parallel?
Show answers (try them yourself first)
  1. 16 ÷ 2 = 3.
  2. 2Rise is 0 ⇒ slope 0.
  3. 3tan 45° = 1.
  4. 4(3 − 7) ÷ (6 − 2) = −4 ÷ 4 = −1.
  5. 5(10 − 1) ÷ (4 + 2) = 9 ÷ 6 = 3/2.
  6. 6Slope of AB is 3, slope of BC is 6 ÷ 2 = 3; yes.
  7. 7(k − 1) ÷ 2 = 4 ⇒ k = 9.
  8. 8Both slopes are 4 ÷ 4 = 1; yes, parallel.