Practice Maths

L43 — Distance, Midpoint and Gradient

Key Terms

Distance formula
d = √[(x2−x1)² + (y2−y1)²] — the straight-line distance between two points on the Cartesian plane.
Midpoint
M = ((x1+x2)/2, (y1+y2)/2) — the average of the x-coordinates and the average of the y-coordinates.
Gradient (slope) m
m = (y2−y1)/(x2−x1) — rise over run; undefined for vertical lines (x1 = x2).
Parallel lines
Lines with equal gradients (m1 = m2) that never intersect.
Perpendicular lines
Lines whose gradients satisfy m1 × m2 = −1; each gradient is the negative reciprocal of the other.
Collinear points
Three or more points are collinear if the gradient between every pair of them is equal — they all lie on the same straight line.

Core formulas

FormulaRuleNotes
Distance d = √[(x2−x1)² + (y2−y1)²] Pythagorean theorem on the coordinate plane
Midpoint M = ((x1+x2)/2, (y1+y2)/2) Average of x-coordinates and y-coordinates
Gradient m = (y2−y1) / (x2−x1) Rise over run; undefined for vertical lines

Gradient relationships

RelationshipConditionExample
Parallel linesm1 = m2Both lines have gradient 3
Perpendicular linesm1 × m2 = −1Gradients 2 and −1/2
Horizontal linem = 0y = 5
Vertical linem undefinedx = 3
x y 1 2 3 4 5 6 7 0 1 2 3 4 run = 6 rise = 3 A(1,1) B(7,4) M(4, 2.5) d = √(6²+3²) = √45 = 3√5 m = 3/6 = 1/2 M = ((1+7)/2, (1+4)/2) = (4, 2.5)
A(1,1) to B(7,4): distance = 3√5, gradient = 1/2, midpoint = (4, 2.5).
Hot Tip: For perpendicular lines, m1 × m2 = −1 — so one gradient is the negative reciprocal of the other. If m1 = 3/4, flip and negate to get m2 = −4/3.

Worked Example 1 — Distance between two points

Find the distance between P(2, −1) and Q(6, 5).

d = √[(6−2)² + (5−(−1))²] = √[16 + 36] = √52 = 2√13 ≈ 7.21.

Worked Example 2 — Midpoint

Find the midpoint of A(−3, 4) and B(7, −2).

M = ((−3+7)/2, (4+(−2))/2) = (4/2, 2/2) = (2, 1).

Worked Example 3 — Gradient

Find the gradient of the line through C(1, 5) and D(4, −1).

m = (−1−5)/(4−1) = −6/3 = −2. The line falls steeply to the right.

Worked Example 4 — Parallel and perpendicular

A line has gradient 3/4. Find the gradient of: (a) a parallel line; (b) a perpendicular line.

(a) Parallel: same gradient = 3/4.

(b) Perpendicular: m × 3/4 = −1 ⇒ m = −4/3.

Worked Example 5 — Finding a missing endpoint

M(3, 1) is the midpoint of AB. A = (1, −3). Find B.

(1+x)/2 = 3 ⇒ x = 5. (−3+y)/2 = 1 ⇒ y = 5. B = (5, 5).

See Answers ➔

  1. Distance formula. Fluency

    • (a) Find the distance between A(0, 0) and B(3, 4).
    • (b) Find the distance between P(1, 2) and Q(4, 6).
    • (c) Find the distance between R(−2, 3) and S(4, −5).
    • (d) A point is 5 units from the origin. If its x-coordinate is 3, find its possible y-coordinates.

  2. Midpoint formula. Fluency

    • (a) Find the midpoint of A(2, 4) and B(8, 10).
    • (b) Find the midpoint of P(−3, 5) and Q(7, −1).
    • (c) M(4, 3) is the midpoint of CD. C = (1, 7). Find D.
    • (d) The midpoint of AB is (0, −2). If A = (3, 4), find B.

  3. Gradient. Fluency

    • (a) Find the gradient of the line through A(1, 3) and B(5, 11).
    • (b) Find the gradient of the line through P(0, 4) and Q(2, 0).
    • (c) A line has gradient −3. What is the gradient of a parallel line?
    • (d) A line has gradient 2. What is the gradient of a perpendicular line?

  4. Parallel and perpendicular. Fluency

    • (a) Line 1 passes through (0, 0) and (2, 6). Line 2 passes through (1, 4) and (3, 10). Are they parallel?
    • (b) Line 1 has gradient 5/2. Find the gradient of a perpendicular line.
    • (c) Are the lines through A(0, 1), B(3, 4) and C(2, 0), D(5, 3) parallel, perpendicular, or neither?
    • (d) A line through the origin is perpendicular to the line through (0, 3) and (4, 1). Find its gradient.

  5. Triangle on the coordinate plane. Understanding

    Triangle ABC has vertices A(0, 0), B(6, 0), and C(3, 4).

    x y 1 2 3 4 5 6 0 1 2 3 A (0,0) B (6,0) C (3,4)
    Triangle ABC with vertices A(0,0), B(6,0), C(3,4). Dashed line is the altitude from C.
    • (a) Find the lengths of all three sides AB, BC, and CA.
    • (b) Find the midpoint of each side.
    • (c) Show that triangle ABC is isosceles.
    • (d) Find the gradient of AB and the gradient of the altitude from C to AB. Verify they are perpendicular.

  6. Collinearity and section formula. Understanding

    • (a) Show that A(1, 1), B(3, 5), and C(5, 9) are collinear (lie on the same line) by checking gradients.
    • (b) The gradient between two points is 3 and the run is 4. What is the rise?
    • (c) P divides AB internally in the ratio 2:1 where A = (0, 0) and B = (6, 9). Find P.
    • (d) A ladder leans against a wall. The base is 2 m from the wall and the top reaches 5 m up. Find the gradient of the ladder and the length of the ladder.

  7. Applications of midpoint. Understanding

    • (a) The vertices of a rectangle are A(0, 0), B(6, 0), C(6, 4), D(0, 4). Show that the diagonals bisect each other (share the same midpoint).
    • (b) A median of a triangle connects a vertex to the midpoint of the opposite side. Triangle PQR has P(2, 6), Q(0, 0), R(8, 0). Find the midpoint of QR and the length of the median from P.
    • (c) A segment has endpoints A(k, 3) and B(2, k). The midpoint is (4, 5). Find k.
    • (d) Two ships start at (0, 0) and (10, 0). They move so that each ship’s position is always the midpoint of the other’s start and finish. Find where they meet.

  8. Gradient and collinearity. Understanding

    • (a) A line has gradient 4. It passes through (2, 3). What is the rise when x increases from 2 to 5?
    • (b) Three points are A(0, 2), B(t, 5), C(8, 14). Find t if A, B, C are collinear.
    • (c) A square has one side from (1, 1) to (5, 3). Find the gradient of the adjacent sides.
    • (d) Two perpendicular lines meet at (3, 4). One passes through (0, 2). Find the equation (in any form) of the other line. Hint: just find its gradient for now.

  9. Coordinate geometry in context. Problem Solving

    A town planner places suburbs on a coordinate grid (units in km): Alpha (0, 0), Beta (8, 0), Gamma (8, 6), Delta (0, 6).

    • (a) Find the perimeter of the quadrilateral formed by the four suburbs.
    • (b) A new school is to be built at the intersection of the two diagonals. Find its coordinates.
    • (c) A bus route runs from Alpha to Gamma. Find the length and gradient of this route.
    • (d) A second bus runs from Beta to Delta. Show this route is perpendicular to the first.

  10. Triangle geometry. Problem Solving

    Triangle ABC has A(2, 1), B(8, 1), C(5, 7).

    • (a) Show that AC = BC (the triangle is isosceles) and find the length of the base AB.
    • (b) Find the coordinates of the centroid G, where G = average of the three vertices.
    • (c) Find the perpendicular bisector of AB (the line perpendicular to AB at its midpoint).
    • (d) Verify that C lies on the perpendicular bisector of AB, and explain why this confirms the triangle is isosceles.