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CBSE · Class 10 · Mathematics

Coordinate Geometry 7.3: Section Formula

5 concepts · 40 practice questions with answers · Free to read

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Mid-point of a segment

From daily life

The meeting point halfway between two friends’ houses on a grid map is the average of their positions.

Simple definition

The mid-point of A(x₁, y₁) and B(x₂, y₂) is M((x₁ + x₂)/2, (y₁ + y₂)/2): the average of the x’s and the average of the y’s.

Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

A(x₁, y₁)B(x₂, y₂)M

In the figure: Equal tick marks show M is halfway between A and B.

Live figure

Mid-point of a segment

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Rule 1: Find the mid-point — Add the two x’s and halve; add the two y’s and halve.

  • Easy: (2, 3) and (8, 7): M = (5, 5).
  • Medium: (−4, 6) and (2, −2): M = (−1, 2).
  • Exam-type: (0, 0) and (6, −8): M = (3, −4).

Rule 2: Find the other end — Put the mid-point in the formula and solve for the unknown end.

  • Easy: M(−2, 8), A(1, 4): B = (−5, 12).
  • Medium: M(3, 1), A(−1, −2): B = (7, 4).
  • Exam-type: M(2, −3), A(5, 0): B = (−1, −6).
Remember

Other end = 2 × mid-point − known end.

Common mistake

Forgetting to halve, giving the sum of the coordinates.

Memory clue

"Mid-point: add and halve, x and y separately."

Practice questions

Easy
  1. 1Find the mid-point of the line segment joining the points A(−9, −5) and B(−23, −19).
  2. 2Find the mid-point of the line segment joining the points A(6, 1) and B(4, −11).
  3. 3Find the mid-point of the line segment joining the points A(9, 1) and B(21, 1).
Medium
  1. 4Find the mid-point of the line segment joining the points A(−1, −3) and B(−15, −3).
  2. 5Find the mid-point of the line segment joining the points A(9, −8) and B(11, −20).
  3. 6The mid-point of the line segment AB is M(−2, 8). If A is (1, 4), find the coordinates of B.
Challenging
  1. 7The mid-point of the line segment AB is M(2, 7). If A is (7, 2), find the coordinates of B.
  2. 8The mid-point of the line segment AB is M(−10, 10). If A is (−7, 4), find the coordinates of B.
Show answers (try them yourself first)
  1. 1(−16, −12)
  2. 2(5, −5)
  3. 3(15, 1)
  4. 4(−8, −3)
  5. 5(10, −14)
  6. 6(−5, 12)
  7. 7(−3, 12)
  8. 8(−13, 16)

Point dividing a segment in a given ratio

From daily life

A signboard placed one-third of the way from the school to the market divides the road in the ratio 1 : 2.

Simple definition

If P divides AB internally in the ratio m : n (AP : PB), then P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). Note the cross pattern: m goes with the far end B.

Formula: P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))

A(x₁, y₁)B(x₂, y₂)Pmn

In the figure: AP : PB = 2 : 3, so P is 2/5 of the way from A to B.

Live figure

Point dividing a segment in a given ratio

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Rule 1: Use the formula — Put m, n and the coordinates carefully; m multiplies the second point.

  • Easy: A(1, 2), B(7, 8), 1 : 2: P = (3, 4).
  • Medium: A(−3, 5), B(9, −7), 1 : 3: P = (0, 2).
  • Exam-type: A(2, −1), B(6, 7), 3 : 1: P = (5, 5).

Rule 2: Fraction-of-the-way shortcut — A ratio m : n means P is m/(m + n) of the way from A to B.

  • Easy: A(0, 0), B(9, 6), ratio 1 : 2: one-third of the way: (3, 2).
  • Medium: A(0, 0), B(10, 5), ratio 2 : 3: two-fifths of the way: (4, 2).
  • Exam-type: Ratio 1 : 1 gives the mid-point.
Remember

The part next to A (m) multiplies B’s coordinate, and the part next to B (n) multiplies A’s.

Common mistake

Swapping m and n, which gives the point from the wrong end.

Memory clue

"m with the far end, n with the near end."

Practice questions

Easy
  1. 1Find the coordinates of the point P which divides the line segment joining A(−3, −4) and B(−3, 6) internally in the ratio 2 : 3 (that is, AP : PB = 2 : 3).
  2. 2Find the coordinates of the point P which divides the line segment joining A(−7, 7) and B(−15, 15) internally in the ratio 3 : 1 (that is, AP : PB = 3 : 1).
  3. 3Find the coordinates of the point P which divides the line segment joining A(1, 1) and B(6, 1) internally in the ratio 2 : 3 (that is, AP : PB = 2 : 3).
Medium
  1. 4Find the coordinates of the point P which divides the line segment joining A(−4, −2) and B(6, −7) internally in the ratio 3 : 2 (that is, AP : PB = 3 : 2).
  2. 5Find the coordinates of the point P which divides the line segment joining A(4, −5) and B(16, −9) internally in the ratio 3 : 1 (that is, AP : PB = 3 : 1).
  3. 6Find the coordinates of the point P which divides the line segment joining A(5, −4) and B(11, −4) internally in the ratio 2 : 1 (that is, AP : PB = 2 : 1).
Challenging
  1. 7Find the coordinates of the point P which divides the line segment joining A(0, −3) and B(−7, −31) internally in the ratio 5 : 2 (that is, AP : PB = 5 : 2).
  2. 8Find the coordinates of the point P which divides the line segment joining A(0, −3) and B(−12, 15) internally in the ratio 5 : 1 (that is, AP : PB = 5 : 1).
Show answers (try them yourself first)
  1. 1(−3, 0)
  2. 2(−13, 13)
  3. 3(3, 1)
  4. 4(2, −5)
  5. 5(13, −8)
  6. 6(9, −4)
  7. 7(−5, −23)
  8. 8(−10, 12)

Finding the ratio, the axis point or an unknown end

From daily life

If you know where a point sits on the road, you can work backwards to find the ratio, or the missing end.

Simple definition

Put the known coordinates in the section formula and solve for m : n or for the unknown end. The x-axis divides a segment where y = 0, giving m : n = −y₁ : y₂; the y-axis where x = 0, giving −x₁ : x₂.

Formula: m : n = −y₁ : y₂

A(x₁, y₁)B(x₂, y₂)y = 0 atthe axism : n =−y₁ : y₂thenfind x

In the figure: x-axis cuts A(2, −3), B(5, 6) in the ratio 3 : 6 = 1 : 2, at the point (3, 0).

Rule 1: Find the ratio — Use the one coordinate that tells you where P is.

  • Easy: P(3, 4) divides A(1, 2), B(7, 8): m : n = 1 : 2.
  • Medium: x-axis divides A(2, −3), B(5, 6): 1 : 2, at (3, 0).
  • Exam-type: y-axis divides A(−2, 4), B(6, −2): 1 : 3, at (0, 2.5).

Rule 2: Find an unknown end — Put the ratio and P in the formula and solve for B.

  • Easy: P(3, 4) divides AB in 1 : 2, A(1, 2): B = (7, 8).
  • Medium: P(4, 6) divides AB in 2 : 1, A(0, 0): B = (6, 9).
  • Exam-type: Check by putting B back in the formula.
Remember

On an axis the other coordinate is 0, so only one coordinate gives the ratio.

Common mistake

Forgetting the minus sign in m : n = −y₁ : y₂; a negative ratio means the point is outside the segment.

Memory clue

"y zero on the x-axis, x zero on the y-axis."

Practice questions

Easy
  1. 1The mid-point of the line segment joining A(−2, y) and B(−8, 1) is M(−5, 2). Find y.
  2. 2The mid-point of the line segment joining A(9, y) and B(−3, −5) is M(3, 0). Find y.
  3. 3In what ratio does the x-axis divide the line segment joining A(5, −6) and B(−7, 7)?
Medium
  1. 4The point P(−3, 25) lies on the line segment joining A(−3, 5) and B(−3, 33). Find the ratio AP : PB.
  2. 5In what ratio does the y-axis divide the line segment joining A(−6, −4) and B(8, −2)?
  3. 6The point P(0, −12) divides the line segment joining A(−6, −6) and B in the ratio 3 : 1 (AP : PB). Find the coordinates of B.
Challenging
  1. 7The point P(−9, 1) divides the line segment joining A(−1, −3) and B in the ratio 2 : 1 (AP : PB). Find the coordinates of B.
  2. 8The point P(−6, −1) divides the line segment joining A(−6, 5) and B in the ratio 2 : 5 (AP : PB). Find the coordinates of B.
Show answers (try them yourself first)
  1. 13
  2. 25
  3. 36 : 7
  4. 45 : 2
  5. 53 : 4
  6. 6(2, −14)
  7. 7(−13, 3)
  8. 8(−6, −16)

Trisection, parallelograms and circles

From daily life

Two poles placed at one-third and two-thirds along a wire divide it into three equal pieces.

Simple definition

The points of trisection divide AB in the ratios 1 : 2 and 2 : 1. In a parallelogram the diagonals share the same mid-point. If AB is a diameter, the centre is its mid-point and the radius is half of AB.

trisection1 : 2, 2 : 1parallelogramsame mid-pointcirclecentre = mid-point

In the figure: Trisection of A(2, −2), B(−7, 4): P = (−1, 0), Q = (−4, 2).

Rule 1: Trisection and parallelogram — Use 1 : 2 and 2 : 1, or equate the two diagonal mid-points.

  • Easy: A(2, −2), B(−7, 4): P (1 : 2) = (−1, 0), Q (2 : 1) = (−4, 2).
  • Medium: A(1, 2), B(4, 3), C(6, 6): fourth vertex D = (3, 5).
  • Exam-type: A(−1, 0), B(2, 1), C(3, 5): D = (0, 4).

Rule 2: Circle with a diameter — Centre = mid-point of the diameter; radius = half the distance AB.

  • Easy: A(2, −3), B(−4, 5): centre (−1, 1), radius 5.
  • Medium: A(−3, 2), B(5, 6): centre (1, 4), radius √20.
  • Exam-type: Centre (2, 3), one end (6, 6): other end (−2, 0).
Remember

In every parallelogram, vertex A + vertex C = vertex B + vertex D.

Common mistake

Pairing the wrong vertices: the diagonals join opposite vertices, not neighbours.

Memory clue

"Diagonals bisect each other: A + C = B + D."

Practice questions

Easy
  1. 1In △ABC, A is (1, 2), B is (0, −6) and C is (8, 8). D is the mid-point of BC. Find the coordinates of D.
  2. 2In △ABC, A is (4, 6), B is (2, 5) and C is (−10, −5). D is the mid-point of BC. Find the coordinates of D.
  3. 3Find the points of trisection of the line segment joining A(−5, 5) and B(−11, 20) (the two points that divide it into three equal parts).
Medium
  1. 4AB is a diameter of a circle, with A(−4, 5) and B(−16, −5). Find the centre of the circle.
  2. 5Find the points of trisection of the line segment joining A(3, 1) and B(6, 7) (the two points that divide it into three equal parts).
  3. 6The centre of a circle is O(1, 0) and one end of a diameter is A(−4, 3). Find the other end of the diameter.
Challenging
  1. 7Three vertices of a parallelogram ABCD, taken in order, are A(5, −5), B(5, 6) and C(1, −6). Find the fourth vertex D.
  2. 8Three vertices of a parallelogram ABCD, taken in order, are A(−5, 5), B(−6, −4) and C(4, 1). Find the fourth vertex D.
Show answers (try them yourself first)
  1. 1(4, 1)
  2. 2(−4, 0)
  3. 3(−7, 10) and (−9, 15)
  4. 4(−10, 0)
  5. 5(4, 3) and (5, 5)
  6. 6(6, −3)
  7. 7(1, −17)
  8. 8(5, 10)

Applications of the section formula

From daily life

Planning a metro station a fixed fraction of the way between two towns, or finding the balance point of a triangular plate, both use this formula.

Simple definition

Turn the story into a segment with a ratio. For the centroid of a triangle use the average of the three vertices: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). In a rhombus the diagonals bisect each other; the mid-point of the hypotenuse of a right triangle is equidistant from all three vertices.

Formula: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

story→ segmentpickm : napplyformulaanswerwith unit

In the figure: Triangle (0, 0), (6, 0), (3, 6): centroid = (9/3, 6/3) = (3, 2).

Rule 1: Real situations — Pick the two ends and the ratio from the story, then use the formula.

  • Easy: A(2, 3), B(10, 7): a point one-fourth of the way (1 : 3): (4, 4).
  • Medium: Triangle (0, 0), (6, 0), (3, 6): centroid (3, 2).
  • Exam-type: Midpoint of (4, 0) and (4, 10) is (4, 5).

Rule 2: Shapes and proofs — Use mid-points of the diagonals to prove properties.

  • Easy: A(0, 0), B(4, 3), C(8, 0), D(4, −3): both diagonals have mid-point (4, 0); with equal sides it is a rhombus.
  • Medium: Right triangle A(0, 0), B(6, 0), C(0, 8): mid-point of BC is (3, 4), at distance 5 from A, half of BC = 10.
  • Exam-type: The same mid-point for two diagonals means a parallelogram.
Remember

Draw the points on a grid first so the ratio is visible.

Common mistake

Using 3 vertices’ sum without dividing by 3 for the centroid.

Memory clue

"Centroid is the average of the three corners."

Practice questions

Easy
  1. 1On a map (1 unit = 1 km), village A is at (5, 3) and village B is at (13, 11). A water tank stands exactly half-way along the straight path AB. Find its coordinates.
  2. 2On a map (1 unit = 1 km), village A is at (8, 7) and village B is at (12, 3). A bridge stands exactly half-way along the straight path AB. Find its coordinates.
  3. 3On a map (1 unit = 1 km), village A is at (9, 5) and village B is at (13, 13). A well stands exactly half-way along the straight path AB. Find its coordinates.
Medium
  1. 4On a map (1 unit = 1 km), town A is at (2, 8) and town B is at (29, 26). A petrol pump lies on the straight road from A to B, at 4/9 of the way from A. Find the coordinates of the petrol pump.
  2. 5On a map (1 unit = 1 km), town A is at (8, 9) and town B is at (−19, 36). A rest house lies on the straight road from A to B, at 4/9 of the way from A. Find the coordinates of the rest house.
  3. 6On a map (1 unit = 1 km), towns A and B are at (8, −9) and (2, −15). A rest house is built at the mid-point of AB. How far is it from the origin of the map (in km)?
Challenging
  1. 7On a map (1 unit = 1 km), towns A and B are at (−12, 6) and (−18, 10). A bus stop is built at the mid-point of AB. How far is it from the origin of the map (in km)?
  2. 8On a map (1 unit = 1 km), towns A and B are at (8, −5) and (16, −13). A petrol pump is built at the mid-point of AB. How far is it from the origin of the map (in km)?
Show answers (try them yourself first)
  1. 1(9, 7)
  2. 2(10, 5)
  3. 3(11, 9)
  4. 4(14, 16)
  5. 5(−4, 21)
  6. 613
  7. 717
  8. 815