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medium difficulty6 min

Section intuition from coordinate graph

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

In one line: The bigger weight pulls the point toward the other endpoint.

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Understand it

  • 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.

Formula and rule

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)).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

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

Method you can reuse

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

Sketch it in your answer

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.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
It is the midpoint, because the point lies exactly halfway between the two endpoints when the two parts are equal.

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.

Don't mix this up

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

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.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    The section formula gives the coordinates of a point that divides a segment internally.

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