C
CraftExam
high importancemedium8 min

Section intuition from coordinate graph

This concept tells where a point lies when it divides a line segment in a given ratio.

Practice This Concept

Learn the concept

Student-friendly explanation

The section formula gives the coordinates of a point that divides a segment internally. It is like a weighted average of the two endpoints. If the ratio changes, the point shifts closer to one endpoint and away from the other. This is very useful when a question gives a ratio instead of a direct midpoint.

How to write this in exams

  1. 1

    Start with the exact idea

    This concept tells where a point lies when it divides a line segment in a given ratio.

  2. 2

    Then show how to use it

    1. Identify endpoints A and B. 2. Read the ratio AP:PB carefully. 3. Put the values into the section formula. 4. Simplify x and y separately. 5. Check whether the point lies closer to the expected endpoint.

  3. 3

    Add one concrete example

    If A(2, 1) and B(8, 7), and P divides AB in the ratio 1:2, then P is (4, 3).

  4. 4

    Avoid this incomplete answer

    A common wrong answer is (5, 4) for the ratio 1:2 case, because students reverse the weights or simply average the coordinates.

Definition

This concept tells where a point lies when it divides a line segment in a given ratio.

Example

If A(2, 1) and B(8, 7), and P divides AB in the ratio 1:2, then P is (4, 3).

Rule to remember

If P divides the line 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)).

Memory hook

The bigger weight pulls the point toward the other endpoint.

Examples and method

Worked example

Let A(2, 1) and B(8, 7). If P divides AB in the ratio 1:2, then P = ((1*8 + 2*2)/3, (1*7 + 2*1)/3) = (4, 3).

Method to apply

1. Identify endpoints A and B. 2. Read the ratio AP:PB carefully. 3. Put the values into the section formula. 4. Simplify x and y separately. 5. Check whether the point lies closer to the expected endpoint.

Diagram support

Draw the segment AB, mark the two parts in the given ratio, and place the point closer to the endpoint with the smaller opposite weight.

How CBSE asks it

Find the coordinates of an internal dividing point, or identify the ratio from the position of a point on a segment.

Avoid common mistakes

Common confusion

Students often reverse the weights or use the midpoint formula even when the ratio is not 1:1.

Common wrong answer

A common wrong answer is (5, 4) for the ratio 1:2 case, because students reverse the weights or simply average the coordinates.

Exam tip

Write the ratio carefully as AP:PB = m:n before substituting into the formula. A small ratio means the point is nearer to that endpoint.

Quick check

A point divides a segment in the ratio 1:1. What special point is it?

It is the midpoint, because the point lies exactly halfway between the two endpoints when the two parts are equal.

Answer writing and exam use

1-mark answer

This concept tells where a point lies when it divides a line segment in a given ratio.

2-mark answer

This concept tells where a point lies when it divides a line segment in a given ratio. If P divides the line 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 A(2, 1) and B(8, 7), and P divides AB in the ratio 1:2, then P is (4, 3).

3-mark answer

The section formula gives the coordinates of a point that divides a segment internally. It is like a weighted average of the two endpoints. If the ratio changes, the point shifts closer to one endpoint and away from the other. This is very useful when a question gives a ratio instead of a direct midpoint. If P divides the line 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)). Let A(2, 1) and B(8, 7). If P divides AB in the ratio 1:2, then P = ((1*8 + 2*2)/3, (1*7 + 2*1)/3) = (4, 3). Find the coordinates of an internal dividing point, or identify the ratio from the position of a point on a segment. A common wrong answer is (5, 4) for the ratio 1:2 case, because students reverse the weights or simply average the coordinates.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept