Class 10 — Mathematics (NCERT)

Coordinate Geometry: Distance Formula and Section Formula

Class 10

✓By the end of this lesson students will be able to recall and understand the Cartesian coordinate system.
✓By the end of this lesson students will be able to apply the distance formula to find the distance between any two points in a plane.
✓By the end of this lesson students will be able to apply the section formula to find the coordinates of a point dividing a line segment internally in a given ratio.
✓By the end of this lesson students will be able to use the midpoint formula as a special case of the section formula.
✓By the end of this lesson students will be able to solve problems involving collinearity and properties of geometric figures using the distance and section formulas.

Key concepts

Cartesian Coordinate System (Review)

In a Cartesian coordinate system, the position of a point in a plane is described by an ordered pair of numbers (x, y), called coordinates. The x-coordinate (abscissa) is the perpendicular distance from the y-axis, and the y-coordinate (ordinate) is the perpendicular distance from the x-axis. The point where the x-axis and y-axis intersect is called the origin, denoted by O(0, 0).

Distance Formula

The distance formula is used to calculate the distance between any two points in a Cartesian plane. If P(x1, y1) and Q(x2, y2) are two points in the plane, the distance between them, denoted by PQ, is given by the formula derived from the Pythagorean theorem.

PQ = √((x2 - x1)² + (y2 - y1)²)

Section Formula (Internal Division)

The section formula helps us find the coordinates of a point that divides a line segment joining two given points in a specific ratio internally. If a point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m1:m2, then the coordinates of P are given by:

x = (m1x2 + m2x1) / (m1 + m2), y = (m1y2 + m2y1) / (m1 + m2)

Midpoint Formula

The midpoint formula is a special case of the section formula where the ratio of division is 1:1 (i.e., m1 = m2 = 1). It gives the coordinates of the midpoint of a line segment.

x = (x1 + x2) / 2, y = (y1 + y2) / 2

Key facts to remember

1The distance between two points P(x1, y1) and Q(x2, y2) is given by d = √((x2 - x1)² + (y2 - y1)²).
2The distance of a point P(x, y) from the origin O(0, 0) is √(x² + y²).
3The coordinates of the point P(x, y) which divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m1:m2 are x = (m1x2 + m2x1) / (m1 + m2) and y = (m1y2 + m2y1) / (m1 + m2).
4The coordinates of the midpoint of the line segment joining A(x1, y1) and B(x2, y2) are ((x1 + x2) / 2, (y1 + y2) / 2).
5Three points A, B, C are collinear if the sum of the distances between two pairs of points is equal to the distance of the third pair (e.g., AB + BC = AC).
6A point on the x-axis has its y-coordinate as 0 (i.e., (x, 0)).
7A point on the y-axis has its x-coordinate as 0 (i.e., (0, y)).

Worked examples

Example 1

Find the distance between the points A(2, 3) and B(4, 1).

ILet the given points be A(x1, y1) = (2, 3) and B(x2, y2) = (4, 1).

IIThe distance formula is d = √((x2 - x1)² + (y2 - y1)²).

IIISubstitute the values into the formula:

IVd = √((4 - 2)² + (1 - 3)²)

Vd = √((2)² + (-2)²)

VId = √(4 + 4)

VIId = √8

VIIId = √(4 × 2)

9d = 2√2

Answer

The distance between the points A(2, 3) and B(4, 1) is 2√2 units.

Example 2

Find the coordinates of the point which divides the line segment joining the points A(4, -3) and B(8, 5) in the ratio 3:1 internally.

ILet the given points be A(x1, y1) = (4, -3) and B(x2, y2) = (8, 5).

IIThe ratio of division is m1:m2 = 3:1.

IIIUsing the section formula for internal division:

IVx = (m1x2 + m2x1) / (m1 + m2)

Vy = (m1y2 + m2y1) / (m1 + m2)

VISubstitute the values for x-coordinate:

VIIx = (3 × 8 + 1 × 4) / (3 + 1)

VIIIx = (24 + 4) / 4

9x = 28 / 4

10x = 7

11Substitute the values for y-coordinate:

12y = (3 × 5 + 1 × (-3)) / (3 + 1)

13y = (15 - 3) / 4

14y = 12 / 4

15y = 3

Answer

The coordinates of the point are (7, 3).

Example 3

In what ratio does the point P(-4, 6) divide the line segment joining the points A(-6, 10) and B(3, -8)?

ILet the point P(-4, 6) divide the line segment joining A(-6, 10) and B(3, -8) in the ratio k:1.

IIHere, (x1, y1) = (-6, 10), (x2, y2) = (3, -8), and (x, y) = (-4, 6).

IIIAlso, m1 = k and m2 = 1.

IVUsing the section formula for the x-coordinate:

Vx = (m1x2 + m2x1) / (m1 + m2)

VI-4 = (k × 3 + 1 × (-6)) / (k + 1)

VII-4(k + 1) = 3k - 6

VIII-4k - 4 = 3k - 6

9-4k - 3k = -6 + 4

10-7k = -2

11k = 2/7

12Alternatively, using the y-coordinate:

13y = (m1y2 + m2y1) / (m1 + m2)

146 = (k × (-8) + 1 × 10) / (k + 1)

156(k + 1) = -8k + 10

166k + 6 = -8k + 10

176k + 8k = 10 - 6

1814k = 4

19k = 4/14

20k = 2/7

Answer

The point P(-4, 6) divides the line segment joining A(-6, 10) and B(3, -8) in the ratio 2:7 internally.

We can use either the x-coordinate or the y-coordinate to find the ratio. Both should yield the same result.

Common mistakes

✗**Sign Errors**: Students often make mistakes with signs, especially when substituting negative coordinates into the formulas or squaring negative numbers.
✗**Incorrect Order of Coordinates**: Swapping x1 with x2 or y1 with y2 inconsistently in the distance formula or section formula.
✗**Mixing up Ratios in Section Formula**: Incorrectly associating m1 with (x1, y1) and m2 with (x2, y2). Remember, m1 is multiplied by (x2, y2) and m2 by (x1, y1).
✗**Forgetting the Denominator**: Omitting the (m1 + m2) term in the denominator of the section formula.
✗**Calculation Errors**: Making arithmetic mistakes during simplification, especially with square roots or fractions.

Exam tips

★**Draw a Diagram**: Always draw a rough sketch of the points and line segments. This helps in visualising the problem and can prevent errors.
★**Write Down the Formula First**: Before substituting values, write down the correct formula. This ensures you are using the right method and helps in recalling the formula.
★**Label Points Clearly**: Clearly label your points as (x1, y1), (x2, y2) and your ratios as m1, m2 at the beginning of your solution. This reduces confusion.
★**Show All Steps**: In an exam, show every step of your calculation. This allows for partial credit even if the final answer is incorrect and makes it easier to trace errors.

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