Practice Maths

Distance and Midpoint

Key Ideas

Key Terms

midpoint
M of a segment joining (x1, y1) and (x2, y2) is: M = ((x1+x2)/2, (y1+y2)/2). Average the x-coordinates, average the y-coordinates.
distance
Between two points is: d = √[(x2−x1)² + (y2−y1)²]. This comes directly from Pythagoras’ theorem.
exact surds
Where appropriate (e.g. 4√2 rather than 5.657…), unless a decimal approximation is asked for.
Hot Tip Before applying the distance formula, sketch the two points on a grid. Draw the horizontal and vertical legs. Then d² = (horizontal length)² + (vertical length)². This confirms you are using Pythagoras correctly.

Distance and midpoint on a coordinate plane

xy 123456 0−1−2−31 A(1,−3) B(5, 1) M(3,−1) run = 4 rise = 4 d = 4√2

Worked Example

Question: Find the midpoint and distance between (1, −3) and (5, 1).

Midpoint:
M = ((1+5)/2, (−3+1)/2) = (6/2, −2/2) = (3, −1)

Distance:
d = √[(5−1)² + (1−(−3))²]
d = √[4² + 4²]
d = √[16 + 16]
d = √32 = √(16×2) = 4√2

Worked Example — Finding the Other Endpoint

Question: The midpoint of AB is M(4, 1). Point A is (1, 5). Find point B.

Step 1: Midpoint formula: (1+Bx)/2 = 4 ⇒ Bx = 7

Step 2: (5+By)/2 = 1 ⇒ By = −3

B = (7, −3)

Where the Distance Formula Comes From

The distance formula is derived directly from Pythagoras' Theorem. If you plot two points A(x1, y1) and B(x2, y2) on a coordinate plane, you can draw a right-angled triangle: the horizontal leg has length |x2 − x1| and the vertical leg has length |y2 − y1|. The distance AB is the hypotenuse. By Pythagoras: d = √((x2 − x1)2 + (y2 − y1)2). The squares remove the need for absolute value signs, since squaring always gives a positive result.

For example, the distance between (1, 2) and (4, 6): d = √((4−1)2 + (6−2)2) = √(9 + 16) = √25 = 5.

Using the Distance Formula

To apply the formula correctly: subtract the x-coordinates, square the result; subtract the y-coordinates, square the result; add the two squared values; take the square root. The order in which you subtract does not matter because you square the result. That is, (x2 − x1)2 = (x1 − x2)2. Leave the answer as an exact surd if the number under the square root is not a perfect square, or round to a specified number of decimal places if asked.

The Midpoint Formula

The midpoint M of a line segment joining (x1, y1) and (x2, y2) is found by averaging the x-coordinates and averaging the y-coordinates: M = ((x1 + x2)/2, (y1 + y2)/2). The logic is simple: the midpoint sits exactly halfway between both endpoints, so you take the average in each direction separately. For example, the midpoint of (2, 5) and (8, 3): M = ((2+8)/2, (5+3)/2) = (5, 4).

Key tip: Midpoint is an average, so it always lies between the two endpoints. If your midpoint x-coordinate is outside the range of the two x-values, you have made an arithmetic error. Also: the midpoint formula gives a point (a coordinate pair), so always write your answer as an ordered pair (x, y), not just a single number.

Applications: Length of a Segment and Centre of a Shape

These formulas are tools for solving geometry problems. Finding the length of a diagonal: if a rectangle has corners at (0,0) and (6,8), the diagonal length is √(62 + 82) = √100 = 10. Finding the centre of a line segment: if you know the endpoints of a diameter of a circle, the midpoint gives the centre. Finding the centre of a rectangle: the midpoint of either diagonal gives the same centre point. These applications appear frequently in coordinate geometry problems.

Working Backwards from the Midpoint

Sometimes you know the midpoint and one endpoint and need to find the other endpoint. Set up the midpoint equation for each coordinate separately. For example, if the midpoint of AB is (3, 7) and A = (1, 4), find B: (1 + xB)/2 = 3 ⇒ 1 + xB = 6 ⇒ xB = 5. Similarly (4 + yB)/2 = 7 ⇒ yB = 10. So B = (5, 10).

Mastery Practice

  1. Find the midpoint of the segment joining each pair of points. Fluency

    1. (2, 4) and (8, 10)
    2. (0, 0) and (6, −4)
    3. (−3, 1) and (5, 7)
    4. (1, −5) and (7, 3)
    5. (−2, −6) and (4, 2)
    6. (3, 9) and (−1, −3)
    7. (0, 5) and (10, −1)
    8. (−4, −2) and (6, 8)

  2. Find the exact distance between each pair of points. Leave answers as simplified surds where appropriate. Fluency

    1. (0, 0) and (3, 4)
    2. (1, 2) and (4, 6)
    3. (2, 3) and (8, 11)
    4. (−1, 1) and (5, 9)
    5. (0, 0) and (5, 5)
    6. (3, 7) and (3, 1)
    7. (−2, 4) and (4, −4)
    8. (1, −3) and (7, 5)

  3. The midpoint M and one endpoint A of a segment are given. Find the other endpoint B. Fluency

    1. M(5, 3), A(2, 1)
    2. M(0, 4), A(−3, 7)
    3. M(6, −1), A(2, 3)
    4. M(−2, 5), A(4, 2)

  4. For each triangle with vertices given, find the exact length of all three sides and classify the triangle as equilateral, isosceles, or scalene. Understanding

    xy 1234 0123 A(0, 0) B(4, 0) C(2, 3) AB AC BC
    1. A(0, 0), B(4, 0), C(2, 3)
    2. P(1, 1), Q(5, 1), R(3, 1+2√3)   (Hint: verify this is equilateral.)
    3. X(−2, −1), Y(2, 2), Z(3, −2)

  5. Using midpoints in geometry. Understanding

    xy 1234567 12345 A(1,1) B(7,1) C(7,5) D(1,5) M(4,3) AC BD
    1. A rectangle has vertices A(1, 1), B(7, 1), C(7, 5), D(1, 5). Show that the diagonals AC and BD have the same midpoint.
    2. M(3, 2) is the midpoint of segment PQ where P(1, k). If the y-coordinate of Q is 6, find k.
    3. Points A(0, 0), B(6, 0), C(6, 4) form a right triangle. Find the midpoint of the hypotenuse and verify that it is equidistant from all three vertices.
    4. The midpoints of the sides of a quadrilateral with vertices (0, 0), (4, 0), (4, 4), (0, 4) are connected. What shape is formed? Support your answer with distance and gradient calculations.

  6. Real-world distance and midpoint problems. Problem Solving

    1. Two towns are at map coordinates A(3, 7) and B(11, 1), where 1 unit = 10 km.
      1. Find the straight-line distance between the towns in kilometres. Give the exact value and a decimal approximation to 1 d.p.
      2. A rest stop will be built at the midpoint between the two towns. Find its coordinates.
    2. A drone flies from launch point L(0, 0) to waypoint W(9, 12) and then to landing point P(18, 0).
      1. Find the total distance flown. Give exact values.
      2. What is the direct distance from L to P?
      3. How much further does the drone travel via W compared to flying directly from L to P?
    3. A circle has centre C(3, −1). Point A(7, 2) lies on the circle. Point B is at the other end of the diameter through A.
      1. Find the radius of the circle.
      2. Find the coordinates of point B.
    4. A student claims that the point (5, 6) is equidistant from (1, 3) and (8, 9). Verify whether this claim is correct by calculating both distances.

  7. Finding perimeters of polygons using the distance formula. Problem Solving

    Recall. The perimeter of a polygon is the sum of all its side lengths. Use the distance formula for each side.
    1. Find the perimeter of the triangle with vertices A(0, 0), B(6, 0) and C(3, 4). Give exact values and a decimal approximation to 1 d.p.
    2. A quadrilateral has vertices P(1, 2), Q(5, 2), R(7, 6) and S(3, 6). Calculate the perimeter and identify the shape (hint: check if opposite sides are equal and if all sides are equal).
    3. Find the perimeter of the triangle with vertices X(−3, 1), Y(3, 1) and Z(0, 5). Is the triangle isosceles? Justify.

  8. Distance and midpoint in context. Problem Solving

    1. A hiker starts at campsite A(2, 5) and walks to a lookout at B(14, 10), then on to a shelter at C(8, 18), where 1 unit = 1 km.
      1. Find the total distance hiked (A to B to C). Give exact values.
      2. What is the straight-line distance from A back to C?
      3. A supply drop point is placed at the midpoint of BC. Find its coordinates.
    2. Point M(4, 7) is the midpoint of segment AB. Point A has coordinates (1, 3).
      1. Find the coordinates of B.
      2. Find the distance AB.

  9. Using distance and midpoint to prove geometric properties. Problem Solving

    Strategy. To prove a quadrilateral is a particular shape, calculate relevant distances and/or midpoints and state what property they demonstrate.
    1. Show that the quadrilateral with vertices A(1, 1), B(5, 1), C(6, 4) and D(2, 4) is a parallelogram. (Hint: show that the diagonals bisect each other by finding both midpoints.)
    2. Points P(0, 0), Q(4, 0) and R(2, 2√3) form a triangle. Calculate all three side lengths and determine the type of triangle. What is the exact perimeter?
    3. The midpoints of the sides of the triangle A(0, 0), B(8, 0) and C(4, 6) are joined to form a smaller triangle. Find the coordinates of the three midpoints, then show that the perimeter of the smaller triangle is exactly half the perimeter of the original.

  10. Finding unknown coordinates from distance or midpoint conditions. Problem Solving

    1. Point A(3, k) is a distance of √25 = 5 units from B(0, 1). Find all possible values of k. Show your working.
    2. The midpoint of segment PQ is M(5, 2). P has coordinates (p, 6) and Q has coordinates (9, q). Find the values of p and q.
    3. Three points are A(0, 0), B(8, 0) and C(x, y). C is equidistant from A and B, and AC = 5 units. Find the two possible positions of C.
    4. A circle has centre O(2, 3) and passes through the point T(6, 6). A chord of the circle has endpoints at T and at another point U on the circle. The midpoint of chord TU is M(5, 5). Find the coordinates of U.
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