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CBSE · Class 10 · Mathematics

Coordinate Geometry 7.2: Distance Formula

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

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Distance between two points

From daily life

To find the straight walking distance between a library and a station on a grid map, make a right triangle and use Pythagoras.

Simple definition

The distance formula gives the distance between A(x₁, y₁) and B(x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The horizontal and vertical differences are the two legs of a right triangle.

Formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]

1234567891012345xyA(2, 1)B(6, 4)43d = 5

In the figure: A(2, 1) and B(6, 4): differences 4 and 3, so d = √(16 + 9) = 5.

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Distance between two points

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Rule 1: Use the formula — Subtract x with x and y with y, square, add, take the square root.

  • Easy: A(2, 1), B(6, 4): d = √(16 + 9) = 5.
  • Medium: (−1, 2) and (4, 14): d = √(25 + 144) = 13.
  • Exam-type: (3, −2) and (−1, 1): d = √(16 + 9) = 5.

Rule 2: Same x or same y — If one coordinate matches, just subtract the other.

  • Easy: (2, 5) and (2, 9): d = 9 − 5 = 4.
  • Medium: (−3, 1) and (4, 1): d = 4 − (−3) = 7.
  • Exam-type: Distance is never negative.
Remember

The order of the points does not matter, because the differences are squared.

Common mistake

Forgetting the square root, or adding instead of subtracting inside the bracket.

Memory clue

"Difference of x, difference of y, square, add, root."

Practice questions

Easy
  1. 1Find the distance between the points (−1, 2) and (−1, 13).
  2. 2Find the distance between the points (−9, 1) and (−5, −9).
  3. 3Find the distance between the points (−2, 2) and (4, 2).
Medium
  1. 4Find the distance between the points (6, 4) and (18, 4).
  2. 5Find the distance between the points (9, 6) and (−9, 7).
  3. 6On a map drawn on a grid (1 unit = 1 km), the temple is at (4, 4) and the market is at (10, 12). Find the straight-line distance between them (in km).
Challenging
  1. 7Find the distance between the points (−2, 5) and (10, 10).
  2. 8Find the distance between the points (−2, 1) and (−8, −7).
Show answers (try them yourself first)
  1. 111
  2. 22√29
  3. 36
  4. 412
  5. 55√13
  6. 610
  7. 713
  8. 810

Distance from the origin

From daily life

The distance of a house at (3, 4) from the town centre (0, 0) is a straight 5 blocks, not 3 + 4 = 7.

Simple definition

The distance of P(x, y) from the origin O(0, 0) is d = √(x² + y²). For a point on an axis the distance is just the number without its sign: |y| for (0, y) and |x| for (x, 0).

Formula: d = √(x² + y²)

1234567891012345xyP(3, 4)O5d = √(x² + y²)

In the figure: P(3, 4): d = √(9 + 16) = 5, the hypotenuse of the 3-4-5 triangle.

Rule 1: Any point — Square both coordinates, add, take the root; signs vanish.

  • Easy: (3, 4): d = 5.
  • Medium: (−5, 12): d = √(25 + 144) = 13.
  • Exam-type: (8, −15): d = √(64 + 225) = 17.

Rule 2: Points on the axes — One coordinate is 0, so the root of a square is the number itself without sign.

  • Easy: (0, −7): d = 7.
  • Medium: (9, 0): d = 9.
  • Exam-type: (0, 0) is at distance 0 from itself.
Remember

The origin is just the second point (0, 0) in the distance formula.

Common mistake

Adding the coordinates (3 + 4) instead of using the root of the sum of squares.

Memory clue

"Square, add, root: the Pythagoras trick from O."

Practice questions

Easy
  1. 1Find the distance of the point (7, −6) from the origin.
  2. 2Find the distance of the point (7, −1) from the origin.
  3. 3Find the distance of the point (−6, −8) from the origin.
Medium
  1. 4Find the distance of the point (7, 5) from the origin.
  2. 5Find the distance of the point (5, 2) from the origin.
  3. 6What is the distance of the point (−4, −3) from the origin?
Challenging
  1. 7What is the distance of the point (−15, 8) from the origin?
  2. 8The point (−12, y) with y > 0 is 15 units from the origin. Find y.
Show answers (try them yourself first)
  1. 1√85
  2. 25√2
  3. 310
  4. 4√74
  5. 5√29
  6. 65
  7. 717
  8. 89

Triangles, collinearity and points on an axis

From daily life

Three bus stops are on one straight road only if the two short gaps add up to the whole gap.

Simple definition

Find the three side lengths with the distance formula. If two shorter lengths add to the longest, the points are collinear. Otherwise compare sides: two equal means isosceles, and a² + b² = c² means right-angled. A point on an axis has one coordinate 0.

find3 sidesadd2 shorter= longestcollineara²+b²=c²right angle

In the figure: A(1, 1), B(3, 3), C(5, 5): AB + BC = 2√2 + 2√2 = 4√2 = AC, so collinear.

Rule 1: Kind of triangle — Compute AB, BC, CA and compare; then test Pythagoras.

  • Easy: A(0, 0), B(6, 0), C(3, 4): AC = BC = 5: isosceles.
  • Medium: A(1, 1), B(4, 1), C(1, 5): 3² + 4² = 5²: right-angled.
  • Exam-type: A(1, 1), B(3, 3), C(5, 5): collinear, no triangle.

Rule 2: Point on an axis — Take (x, 0) or (0, y) and make the two distances equal.

  • Easy: On the x-axis, equidistant from (2, −5) and (−2, 9): (−7, 0).
  • Medium: On the y-axis, equidistant from (6, 5) and (−4, 3): (0, 9).
  • Exam-type: Check: both distances for (−7, 0) are √106.
Remember

Compare squares of distances to avoid roots until the end.

Common mistake

Rounding the roots to decimals before adding them; keep the roots exact (like 2√2) when testing collinearity.

Memory clue

"Short plus short equals long: one straight line."

Practice questions

Easy
  1. 1The vertices of a triangle are H(0, 4), V(3, 9) and S(−2, 10). Is HV = HS?
  2. 2The vertices of a triangle are Y(−5, 4), P(0, 8) and G(0, 0). Is YP = YG?
  3. 3The vertices of a triangle are X(1, −6), Q(3, −4) and A(2, −7). Is XQ = XA?
Medium
  1. 4Are the points (0, −2), (1, 0) and (3, 2) collinear?
  2. 5Are the points (0, −6), (5, 3) and (2, −1) collinear?
  3. 6Find the point on the x-axis which is equidistant from (2, 2) and (7, 7).
Challenging
  1. 7Find the point on the y-axis which is equidistant from (−5, −6) and (1, 6).
  2. 8Find the point on the x-axis which is equidistant from (−3, 3) and (−5, 5).
Show answers (try them yourself first)
  1. 1No
  2. 2Yes
  3. 3No
  4. 4No
  5. 5No
  6. 6(9, 0)
  7. 7(0, −1)
  8. 8(−8, 0)

Finding an unknown coordinate

From daily life

When a pole’s position has one unknown coordinate but its distance from another pole is known, the formula becomes an equation.

Simple definition

Put the given distance in the formula, square both sides to remove the root, and solve for the unknown. For an equidistant point set the two squared distances equal. Check each answer.

putdistancesquareboth sidessolvecheck

In the figure: (2, −3) and (10, y) at distance 10: 64 + (y + 3)² = 100, so y = 3 or −9.

Rule 1: Given distance — Square both sides, solve, and keep both roots if both work.

  • Easy: (2, −3) and (10, y), distance 10: y = 3 or −9.
  • Medium: P(x, 2) and Q(9, 8), distance 10: x = 1 or 17.
  • Exam-type: Both values give a valid point, so two answers.

Rule 2: Equidistant point — Set PA² = PB²; the squares of the unknowns cancel.

  • Easy: (x, y) equidistant from (7, 1) and (3, 5): x − y = 2.
  • Medium: Check: (4, 2) gives √10 to both points.
  • Exam-type: The condition is a line, the perpendicular bisector.
Remember

Squaring both sides is safe here because both sides are non-negative.

Common mistake

Writing (y + 3)² as y² + 9 and losing the middle term.

Memory clue

"Square, expand fully, solve the quadratic."

Practice questions

Easy
  1. 1The distance between the points (k, −8) and (8, −8) is 9. If k < 8, find k.
  2. 2The distance between the points (k, −6) and (−3, −6) is 9. If k > −3, find k.
  3. 3The distance between the points (−4, y) and (−16, 6) is 20. Find the value(s) of y.
Medium
  1. 4The distance between the points (−7, y) and (5, 5) is 15. Find the value(s) of y.
  2. 5The distance between the points (0, y) and (12, 2) is 13. Find the value(s) of y.
  3. 6The point P(x, −5) is equidistant from A(2, 6) and B(1, −5). Find x.
Challenging
  1. 7The point P(x, −3) is equidistant from A(5, −2) and B(1, 0). Find x.
  2. 8The point P(x, 1) is equidistant from A(4, 2) and B(0, −4). Find x.
Show answers (try them yourself first)
  1. 1−1
  2. 26
  3. 3−10 or 22
  4. 4−4 or 14
  5. 5−3 or 7
  6. 662
  7. 72
  8. 8−1

Quadrilaterals and circles

From daily life

Surveyors decide whether a plot is a square, a rhombus or a circle just by measuring distances between corner points.

Simple definition

Find all four sides and both diagonals: equal sides + equal diagonals = square; equal sides, unequal diagonals = rhombus; opposite sides equal = parallelogram. A point is on a circle if its distance from the centre equals the radius, inside if less, outside if more.

4 sidesequal?diagonalsequal?oppositesides equal?type

In the figure: A(0, 0), B(4, 3), C(8, 0), D(4, −3): all sides 5, diagonals 8 and 6, so a rhombus.

Rule 1: Name the quadrilateral — Compare the sides first, then the diagonals.

  • Easy: A(0, 0), B(4, 0), C(4, 4), D(0, 4): sides 4, diagonals 4√2: square.
  • Medium: A(0, 0), B(4, 3), C(8, 0), D(4, −3): rhombus.
  • Exam-type: A(0, 0), B(5, 1), C(7, 4), D(2, 3): opposite sides √26 and √13: parallelogram.

Rule 2: Point and circle — Compare the distance from the centre with the radius.

  • Easy: Centre (0, 0), radius 5: (3, 4) lies on the circle.
  • Medium: Same circle: (4, 4) is outside (√32 > 5); (2, 1) is inside (√5 < 5).
  • Exam-type: Centre (2, −3), radius 5: (5, 1) and (6, −6) are on the circle.
Remember

A square needs both equal sides and equal diagonals; equal sides alone give a rhombus.

Common mistake

Calling a shape a square because all four sides are equal, without checking diagonals.

Memory clue

"Sides, then diagonals, then name."

Practice questions

Easy
  1. 1A circle has centre (8, −6) and passes through the point (16, −6). Find its radius.
  2. 2A circle has centre (−8, −1) and passes through the point (−22, −1). Find its radius.
  3. 3A circle has centre (7, 1) and passes through the point (−7, 1). Find its radius.
Medium
  1. 4A circle has centre (6, 3) and passes through the point (6, −4). Find its radius.
  2. 5A circle with centre (−5, −2) passes through the point (5, 0). Find its radius.
  3. 6A circle with centre (−4, −7) passes through the point (−7, 5). Find its radius.
Challenging
  1. 7A circle has centre O(5, −2) and radius 9 units. Where does the point P(11, 6) lie?
  2. 8The points A(−4, −2), B(−8, 2), C(−12, −2) and D(−8, −6) are the vertices of a quadrilateral ABCD taken in order. What is ABCD?
Show answers (try them yourself first)
  1. 18
  2. 214
  3. 314
  4. 47
  5. 52√26
  6. 63√17
  7. 7Outside the circle
  8. 8Square