Section formula | ICSE Class 10 Maths Notes

This note covers coordinates, internal division of a line segment, the section formula and its derivation, midpoint calculations, unknown ratios, trisection, division by coordinate axes, parallelogram applications, and the coordinates of the centroid of a triangle.

What does it mean to divide a line segment internally?

A line segment is the straight part of a line between two endpoints. Let those endpoints be A and B. If a point P lies between them on the segment, P divides AB internally. The notation AB also represents the length of that segment when used in a calculation.

The ratio AP : PB compares the length from A to P with the length from P to B, in that order. Write AP : PB = m : n, where m and n are positive numbers representing the relative lengths of the two parts.

How are points represented?

An ordered pair (x, y) gives the coordinates of a point. The horizontal coordinate x is its abscissa; the vertical coordinate y is its ordinate. Their signs indicate the directions from the coordinate axes. The horizontal x-axis and vertical y-axis meet at the origin, O(0, 0).

Write A(x₁, y₁), B(x₂, y₂) and P(x, y). Here x₁ and y₁ belong to A, x₂ and y₂ belong to B, and x and y belong to P. The subscripts 1 and 2 identify endpoints; they do not indicate multiplication.

Definition: Internal division means that P lies on the segment joining A and B, between its endpoints. The ratio AP : PB specifies how that segment is split.

Read the ratio before calculating. If PB is twice AP, then AP : PB = 1 : 2. The first number belongs to the segment named first. Reversing the endpoint order changes the ratio to PB : PA = 2 : 1.

Collinear points lie on one straight line. Internal division requires collinearity as well as the correct ratio of distances. A distance ratio by itself does not establish that a point lies on the joining segment.

What is the section formula and how is it used?

Result: Coordinates of an internal division point

For the endpoints and symbols defined above, suppose AP : PB = m : n. The section formula gives the coordinates of P by combining the coordinates of A and B with the two ratio numbers.

x = (mx₂ + nx₁)/(m + n)

y = (my₂ + ny₁)/(m + n)

The numerator, which is the expression above the fraction line, combines corresponding coordinates. The denominator, below the fraction line, is the sum m + n. Use the same denominator for both coordinates. Since m and n are positive, that sum is not zero.

Notice the cross-pairing: m multiplies B's coordinates, while n multiplies A's coordinates. The ratio number attached to AP therefore multiplies the coordinates of B. Mixing this order gives a different division point when the two parts are unequal.

1. Name the endpoints A and B, preserving the order used in the question.
2. Write AP : PB explicitly and identify the two ratio numbers.
3. Substitute both x-coordinates into the first formula and both y-coordinates into the second.
4. Simplify the fractions and state the answer as an ordered pair.

Worked example 1. Find P dividing the segment from A(4, −3) to B(8, 5) internally in the ratio AP : PB = 3 : 1.

Answer: Here m = 3 and n = 1. Thus x = [3 × 8 + 1 × 4]/(3 + 1) = 28/4 = 7. Also y = [3 × 5 + 1 × (−3)]/4 = 12/4 = 3. Therefore P = (7, 3).

The brackets around −3 preserve its sign during multiplication. The point lies nearer B because PB is the shorter part. This is a useful check on the endpoint order, although the exact coordinates still come from the calculation.

Why does the section formula work?

The formula follows from similar triangles, whose corresponding angles are equal and corresponding sides are proportional. The AA similarity criterion establishes similarity when two angles of one triangle equal two angles of another.

In this construction, R, S and T are the feet of perpendiculars from A, P and B to the x-axis. Q lies on PS at A's height; C lies on BT at P's height. A perpendicular meets another line at a right angle.

A, P and B lie along a rising segment. Vertical lines from them meet the horizontal axis at R, S and T. Horizontal segments AQ and PC meet the vertical lines through P and B. The diagram labels the two parts AP and PB with the ratio numbers.

See Fig. 7.10 in your NCERT textbook

How does similarity produce the x-coordinate?

1. The right triangles PAQ and BPC are similar by AA. Consequently PA/PB = AQ/PC = PQ/BC.
2. In the illustrated arrangement, AQ = x − x₁ and PC = x₂ − x. Substituting AP : PB = m : n gives m/n = (x − x₁)/(x₂ − x).
3. Cross-multiply to obtain m(x₂ − x) = n(x − x₁). Expanding gives mx₂ − mx = nx − nx₁.
4. Collect the terms containing x: mx₂ + nx₁ = (m + n)x. Dividing by m + n gives the x-coordinate formula.

For the vertical sides, PQ = y − y₁ and BC = y₂ − y in the illustrated arrangement. Substituting these expressions into the same proportionality gives m(y₂ − y) = n(y − y₁). Rearrangement gives y = (my₂ + ny₁)/(m + n).

The diagram explains why both coordinates use the same ratio. They describe the same point on the same segment. The derivation uses a convenient sloping arrangement; the section formula itself applies to internal division of a segment with any orientation.

How can a worded distance condition become a section ratio?

A worded problem may compare the two parts without writing a ratio. Translate the statement into named segment lengths before substituting coordinates. Identify the endpoints, the point between them, and which of the two distances is larger.

How is the relay tower located?

Worked example 2. A town A is the origin (0, 0). Town B is 36 kilometres (km) east and 15 km north of A, so B has coordinates (36, 15), using 1 km as one unit on each axis. A relay tower P lies on AB with PB twice AP. Find P.

Answer: PB = 2AP means AP : PB = 1 : 2. Therefore x = (1 × 36 + 2 × 0)/3 = 12 and y = (1 × 15 + 2 × 0)/3 = 5. The tower is at P(12, 5), corresponding to 12 km east and 5 km north of A.

The phrase “distance from B is twice its distance from A” makes PB the longer segment. It does not make the first ratio number larger. Writing AP : PB before reading off the numbers prevents that reversal.

How does a fraction of the whole differ from a ratio of parts?

Worked example 3. Let A(−2, −2) and B(2, −4) be endpoints. P lies on AB and AP = 3AB/7. Find P.

Answer: The remaining part is PB = 4AB/7, so AP : PB = 3 : 4. Hence x = [3 × 2 + 4 × (−2)]/7 = −2/7 and y = [3 × (−4) + 4 × (−2)]/7 = −20/7. Thus P = (−2/7, −20/7).

Part-to-whole and part-to-part comparisons have different denominators. In this example, 3/7 compares AP with AB; the ratio 3 : 4 compares AP with PB. The section formula requires the second comparison. Subtracting the given fraction from the whole supplies the missing part.

How does the midpoint formula follow from internal division?

Result: Coordinates of the midpoint

A midpoint divides a segment into two equal lengths. If M is the midpoint of AB, then AM = MB and AM : MB = 1 : 1. Substitute equal ratio numbers into the section formula.

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Each coordinate is the arithmetic mean, or sum divided by the number of values, of the corresponding endpoint coordinates. Add the two x-coordinates together, then the two y-coordinates. Do not combine the x-coordinate of one endpoint with the y-coordinate of the other.

The midpoint calculation retains negative signs. Equal division concerns segment lengths; it does not mean that the endpoint coordinates must be positive or that the midpoint must be the origin. The coordinate sums determine the actual position.

How can a midpoint reveal a missing endpoint?

Worked example 4. AB is a diameter of a circle with centre (2, −3). Given B(1, 4), find A.

Answer: A diameter is a segment through the centre with endpoints on the circle, so the centre is its midpoint. Let A have unknown coordinates (a, b). Then (a + 1)/2 = 2 and (b + 4)/2 = −3. Thus a = 3 and b = −10, giving A(3, −10).

Here a and b denote the unknown x-coordinate and y-coordinate of A. Multiplying each midpoint equation by 2 gives the sum of the endpoint coordinates. Subtracting the known coordinate then isolates the missing one.

Check by substitution: the midpoint of (3, −10) and (1, 4) is ((3 + 1)/2, (−10 + 4)/2) = (2, −3). This recovers both supplied centre coordinates and checks the signs independently.

How can the section formula find an unknown ratio?

If both endpoints and the division point are known, the coordinates become data and the ratio becomes the unknown. Write the ratio as k : 1, where k is the positive quotient AP/PB. This reduces the calculation to one unknown number.

With the earlier endpoint notation, the section formula becomes x = (kx₂ + x₁)/(k + 1) and y = (ky₂ + y₁)/(k + 1). Equate one expression to the known coordinate of P, solve for k, and check the other coordinate.

Worked example 5. Find the ratio AP : PB for A(−6, 10), P(−4, 6) and B(3, −8), where P divides AB internally.

Answer: Put AP : PB = k : 1. Using the x-coordinate, −4 = (3k − 6)/(k + 1). Thus −4k − 4 = 3k − 6, giving 7k = 2 and k = 2/7. Therefore AP : PB = 2 : 7. The y-coordinate check is [2 × (−8) + 7 × 10]/9 = 54/9 = 6.

Why check the second coordinate?

An equality of ordered pairs requires equality in both positions. Solving the x-coordinate equation determines a candidate ratio; substituting it in the y-coordinate equation checks that it produces the whole supplied point.

The ratio (2/7) : 1 is equivalent to 2 : 7 because multiplying both entries by 7 leaves the comparison unchanged. Report a simplified ratio with its segment order, rather than leaving k as an unexplained fraction.

Note: The ratio can also be found from the lengths PA and PB, provided you know that A, P and B are collinear. Keep this condition when using a distance-based method.

The endpoint order matters during checking too. A result written as BP : PA would be 7 : 2 for the same three points. That describes the reverse comparison, rather than a different geometric position.

How are trisection points found?

Trisection means dividing a segment into three equal parts. Two internal points are needed. Let them be P and Q in the order A, P, Q, B, so AP = PQ = QB.

For P, the segment PB contains two equal parts while AP contains one. Thus AP : PB = 1 : 2. For Q, AQ contains two equal parts while QB contains one. Thus AQ : QB = 2 : 1.

The schematic shows A, P, Q and B in that order on a straight segment. The endpoint labels are A(2, −2) and B(−7, 4). P and Q mark the three equal parts.

See Fig. 7.11 in your NCERT textbook

How are the two ratios applied?

Worked example 6. Find both trisection points of the segment joining A(2, −2) and B(−7, 4).

Answer: For P, use 1 : 2: x = [1 × (−7) + 2 × 2]/3 = −1 and y = [1 × 4 + 2 × (−2)]/3 = 0. For Q, use 2 : 1: x = [2 × (−7) + 1 × 2]/3 = −4 and y = [2 × 4 + 1 × (−2)]/3 = 2. Hence P(−1, 0) and Q(−4, 2) are the required points.

Both applications use A and B as the endpoints. The difference lies in the ratio. Do not use 1 : 3 for the first point: that compares the first part with the entire segment, whereas the formula compares the two parts on either side of the point.

There is also a midpoint check. Once P is known, Q is the midpoint of PB because PQ = QB. The midpoint of P(−1, 0) and B(−7, 4) is ((−1 − 7)/2, (0 + 4)/2) = (−4, 2), agreeing with the section calculation.

A complete trisection answer names both points and keeps their order clear. The first is reached from A after one equal part; the second is reached after two. One coordinate pair alone leaves half the required answer missing.

How do coordinate axes determine a division point?

A point on the y-axis has x-coordinate zero and is written (0, y). A point on the x-axis has y-coordinate zero and is written (x, 0). These conditions supply an equation when the ratio of division is unknown.

Let the intersection point be P, meaning the point shared by the segment and the specified axis. Set AP : PB = k : 1, use the zero coordinate to find k, then substitute into the other coordinate expression.

Worked example 7. Find the ratio in which the y-axis divides the segment joining A(5, −6) and B(−1, −4), and find the point of intersection.

Answer: Let AP : PB = k : 1. The x-coordinate of P is (−k + 5)/(k + 1). Since P lies on the y-axis, this equals 0, giving k = 5. Hence the ratio is 5 : 1. Its y-coordinate is [5 × (−4) − 6]/6 = −26/6 = −13/3. Therefore P = (0, −13/3).

How does the condition change for the x-axis?

Worked example 8. Find the ratio in which the x-axis divides the segment joining A(1, −5) and B(−4, 5), and find the division point.

Answer: Put AP : PB = k : 1. The y-coordinate is (5k − 5)/(k + 1). Setting it equal to 0 gives k = 1, so AP : PB = 1 : 1. The x-coordinate is (−4 + 1)/2 = −3/2. Thus P = (−3/2, 0).

These examples have an internal crossing of the specified axis. The positive value of k agrees with internal division. The point's final ordered pair should visibly satisfy the relevant axis condition, which gives a quick check independent of the ratio algebra.

Do not stop after obtaining k if the question also asks for the intersection. The ratio describes how the segment is divided; the ordered pair locates the point. They are two related but distinct pieces of information.

How is the midpoint formula used in a parallelogram?

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A diagonal joins two opposite vertices, where a vertex is a corner. The diagonals of a parallelogram bisect each other: each cuts the other into two equal parts.

For vertices A, B, C and D taken in order around the boundary, the diagonals are AC and BD. Their common intersection is the midpoint of each. Calculate these two midpoints and equate their corresponding coordinates.

Worked example 9. A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order. Here p is the unknown x-coordinate of D. Find p.

Answer: The midpoint of AC is ((6 + 9)/2, (1 + 4)/2) = (15/2, 5/2). The midpoint of BD is ((8 + p)/2, (2 + 3)/2) = ((8 + p)/2, 5/2). Equating x-coordinates gives 15/2 = (8 + p)/2, so p = 7.

What does “taken in order” tell you?

It identifies which vertices are adjacent and which are opposite. This matters because the midpoint property concerns the diagonals, not adjacent sides. For the given order, pairing A with B and C with D would apply the formula to the wrong segments.

The two y-coordinates already agree at 5/2. They confirm consistency but do not determine p, which occurs in the x-coordinate equation. Match x with x and y with y when comparing midpoint expressions.

After finding p, substitute D(7, 3) back into the midpoint of BD. It becomes ((8 + 7)/2, (2 + 3)/2) = (15/2, 5/2), matching AC. The check tests the geometric condition used to construct the equation.

How are the coordinates of a triangle's centroid obtained?

A median of a triangle joins a vertex to the midpoint of the opposite side. The centroid is the common point of its three medians. It divides each median internally in the ratio 2 : 1, measured from the vertex towards the opposite side's midpoint.

Result: Coordinates of the centroid

Let the triangle's vertices be A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃), where x₃ and y₃ are C's coordinates. Denote the centroid by G. Its coordinates are the separate arithmetic means of the three vertex coordinates.

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

How does section division give this result?

1. Let D be the midpoint of BC. Its coordinates are ((x₂ + x₃)/2, (y₂ + y₃)/2).
2. The median from A is AD. Take the point on AD dividing it in the ratio AG : GD = 2 : 1.
3. The section formula gives its x-coordinate as [2 × (x₂ + x₃)/2 + x₁]/3 = (x₁ + x₂ + x₃)/3.
4. Its y-coordinate is [2 × (y₂ + y₃)/2 + y₁]/3 = (y₁ + y₂ + y₃)/3. Repeating from either other vertex gives the same coordinate pair.

This shared pair lies on all three medians, establishing the centroid result and the direction of the 2 : 1 division. The larger part runs from the vertex to the centroid. The smaller part runs from the centroid to the side's midpoint.

For a missing vertex, reverse the centroid equations separately. Multiply each centroid coordinate by 3, then subtract the corresponding coordinates of the two known vertices. This is the same rearrangement idea used to recover an endpoint from a midpoint.

The centroid formula does not average a mixture of horizontal and vertical coordinates. Keeping the two sums separate preserves the ordered-pair structure, while the denominator 3 reflects the three vertices included in each sum.

Glossary

• Coordinates — An ordered pair of numbers locating a point relative to the two coordinate axes.
• Abscissa — The x-coordinate of a point, giving its horizontal position relative to the y-axis.
• Ordinate — The y-coordinate of a point, giving its vertical position relative to the x-axis.
• Internal division — Division of a segment by a point lying between its two endpoints on that segment.
• Section formula — A formula giving a division point's coordinates from endpoint coordinates and the ratio of division.
• Midpoint — The point on a line segment that divides it into two equal lengths.
• Collinear points — Points lying on the same straight line, a condition needed for segment division.
• Trisection — Division of a line segment into three equal parts using two internal points.
• Median — A segment joining a triangle's vertex to the midpoint of its opposite side.
• Centroid — The common point of a triangle's three medians, dividing each in the ratio two to one from its vertex.
• Diagonal — A segment joining opposite vertices of a quadrilateral, such as a parallelogram.
• Bisect — To divide a line segment into two parts of equal length.

Common errors and misconceptions

• Misconception: AP : PB = m : n means m multiplies A's coordinates. Correct: In the internal section formula, m multiplies B's coordinates and n multiplies A's coordinates.
• Misconception: Internal division uses the difference of the ratio numbers as denominator. Correct: Both coordinate expressions use their sum, m + n.
• Misconception: AP = 3AB/7 means AP : PB = 3 : 7. Correct: PB is the remaining 4AB/7, so the required ratio is 3 : 4.
• Misconception: A point on the y-axis has y-coordinate zero. Correct: Its x-coordinate is zero. On the x-axis, the y-coordinate is zero.
• Misconception: Trisection requires a single midpoint calculation. Correct: There are two points, dividing the full segment in ratios 1 : 2 and 2 : 1.
• Misconception: A distance ratio alone proves internal division. Correct: The point must also lie on the joining segment; retain the condition of collinearity.
• Misconception: The centroid is halfway along each median. Correct: It divides each median in the ratio 2 : 1 from the vertex; its coordinates average three vertices.

Exam-style questions with model answers

Q1. Find P dividing A(4, −3) and B(8, 5) internally with AP : PB = 3 : 1. [2 marks]

1. Apply the section formula to the x-coordinates: x = (3 × 8 + 1 × 4)/4 = 7.
2. Similarly, y = [3 × 5 + 1 × (−3)]/4 = 3. Therefore the required point is P(7, 3).

Q2. AB is a diameter of a circle with centre (2, −3). If B is (1, 4), find A using the midpoint formula. [3 marks]

1. Let A have unknown coordinates (a, b). The centre of the circle is the midpoint of its diameter AB, so its coordinates equal ((a + 1)/2, (b + 4)/2).
2. Equating x-coordinates gives (a + 1)/2 = 2. Hence a + 1 = 4 and a = 3.
3. Equating y-coordinates gives (b + 4)/2 = −3. Hence b + 4 = −6 and b = −10. Thus A is (3, −10).

Q3. P(−4, 6) divides A(−6, 10) and B(3, −8) internally. Find AP : PB and check both coordinates. [4 marks]

1. Let AP : PB = k : 1, where k is positive. The section formula gives −4 = (3k − 6)/(k + 1).
2. Cross-multiplication gives −4k − 4 = 3k − 6. Therefore 7k = 2, so k = 2/7 and AP : PB = 2 : 7.
3. Substitute this ratio into the x-coordinate expression: [2 × 3 + 7 × (−6)]/9 = −36/9 = −4.
4. The y-coordinate expression gives [2 × (−8) + 7 × 10]/9 = 54/9 = 6. Both coordinates reproduce P(−4, 6), confirming the ratio.

Q4. Find both trisection points P and Q of A(2, −2) and B(−7, 4), in the order A, P, Q, B. Check Q using a midpoint calculation. [5 marks]

1. Trisection means AP = PQ = QB. The first point P therefore divides the full segment AB in the ratio AP : PB = 1 : 2.
2. Apply the section formula: P has x-coordinate [−7 + 2 × 2]/3 = −1 and y-coordinate [4 + 2 × (−2)]/3 = 0. Thus P is (−1, 0).
3. The second point Q divides AB in the ratio AQ : QB = 2 : 1, because AQ contains two of the three equal parts.
4. Its coordinates are ([2 × (−7) + 2]/3, [2 × 4 − 2]/3) = (−4, 2). Thus the two required points are P(−1, 0) and Q(−4, 2).
5. Since PQ = QB, Q is the midpoint of PB. That midpoint is ((−1 − 7)/2, (0 + 4)/2) = (−4, 2), confirming the second point.

Q5. Find the ratio in which the y-axis divides A(5, −6) and B(−1, −4), and find the intersection point. [3 marks]

1. Let P be the intersection and AP : PB = k : 1. Since P lies on the y-axis, its x-coordinate is zero: (−k + 5)/(k + 1) = 0.
2. The numerator must be zero, so −k + 5 = 0 and k = 5. The ratio is therefore AP : PB = 5 : 1.
3. Substitution into the y-coordinate formula gives y = [5 × (−4) − 6]/6 = −13/3. Hence the point of intersection is P(0, −13/3).

Q6. A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are vertices of a parallelogram taken in order. Find the unknown coordinate p. [3 marks]

1. The diagonals AC and BD bisect each other, so their midpoints coincide. The midpoint of AC is ((6 + 9)/2, (1 + 4)/2) = (15/2, 5/2).
2. The midpoint of BD is ((8 + p)/2, (2 + 3)/2) = ((8 + p)/2, 5/2). Its y-coordinate already agrees with that of AC's midpoint.
3. Equating x-coordinates gives (8 + p)/2 = 15/2. Therefore 8 + p = 15 and p = 7, giving D(7, 3).

Q7. A triangle has vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃), where each ordered pair gives that vertex's coordinates. D is the midpoint of BC. The centroid G divides AD with AG : GD = 2 : 1. Derive G's coordinates. [5 marks]

1. Since D is the midpoint of BC, its coordinates are ((x₂ + x₃)/2, (y₂ + y₃)/2), found by averaging the corresponding coordinates of B and C.
2. Apply internal section division to the segment from A to D with ratio 2 : 1. The coefficient 2 multiplies D's coordinates; the coefficient 1 multiplies A's coordinates.
3. The x-coordinate of G is [2 × (x₂ + x₃)/2 + x₁]/3. Cancelling the factor 2 in the first term gives (x₁ + x₂ + x₃)/3.
4. The same calculation for the y-coordinate gives [2 × (y₂ + y₃)/2 + y₁]/3 = (y₁ + y₂ + y₃)/3.
5. Thus G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). Each coordinate is the arithmetic mean of the corresponding coordinates of the three vertices.

Key takeaways

• Internal division requires a point between the endpoints on their joining segment, with a specified ratio of the two parts.
• For AP : PB = m : n, multiply B's coordinates by m and A's coordinates by n, then divide by m + n.
• The midpoint formula is the equal-ratio case of the section formula, averaging corresponding coordinates of the two endpoints.
• Translate a fraction of the whole segment into a ratio of its two parts before applying internal division.
• For an unknown ratio, use one coordinate to solve the equation and the other to check the supplied point.
• Trisection needs two points, using ratios 1 : 2 and 2 : 1 with the same endpoint order.
• On the y-axis set the x-coordinate to zero; on the x-axis set the y-coordinate to zero.
• A triangle's centroid averages the three vertex coordinates and divides each median two to one from the vertex.

Test yourself

For A(x₁, y₁), B(x₂, y₂) and AP : PB = m : n, which endpoint's coordinates are multiplied by m?

The coordinates of B are multiplied by m; those of A are multiplied by n.

If P is on AB and PB is twice AP, what is AP : PB?

The ratio is 1 : 2 because the second named segment is twice the first.

If P lies on AB and AP = 3AB/7, what ratio belongs in the section formula?

Use AP : PB = 3 : 4, because PB is the remaining four-sevenths of AB.

What ratio makes the section formula become the midpoint formula?

The ratio 1 : 1 gives equal parts and averages the corresponding endpoint coordinates.

Which coordinate is zero at an intersection with the y-axis?

The x-coordinate is zero, so the point has the form (0, y).

For trisection points in the order A, P, Q, B, what are AP : PB and AQ : QB?

They are 1 : 2 and 2 : 1 respectively, each comparing parts of the full segment.

What condition must be known before using PA/PB as a section ratio?

A, P and B must be collinear; for internal division P must lie between the endpoints.

If D is the midpoint of BC and G is the centroid of triangle ABC, what is AG : GD?

It is 2 : 1, measured from vertex A towards the opposite side's midpoint D.