Hard GCSE Coordinates Questions

Challenging, exam-style GCSE Coordinates questions with worked solutions. Stretch yourself on the hardest distance, Pythagoras, surd form, reasoning problems.

distancePythagorassurd formreasoningparallelogramvectors
GCSE Foundation34 questionsStep-by-step solutions
Question 1
5 markschallenging
A=(1,2)A=(1, 2), B=(1,7)B=(1, 7), C=(10,7)C=(10, 7) and D=(10,2)D=(10, 2) are the vertices of rectangle ABCDABCD. Work out its area and perimeter.
Show worked solution

Worked solution

  1. List the four vertices

    A=(1,2),  B=(1,7),  C=(10,7),  D=(10,2)A=(1, 2),\; B=(1, 7),\; C=(10, 7),\; D=(10, 2)

    We confirm ABCD is a rectangle and find its area and perimeter.

  2. Vector AB

    AB⃗=(0,5)\vec{AB} = (0, 5)

    Find the direction and length of side AB.

  3. Length of AB

    AB=(0)2+(5)2=5AB = \sqrt{(0)^2+(5)^2} = 5

    Side AB is 5 units long.

  4. Vector BC

    BC⃗=(9,0)\vec{BC} = (9, 0)

    Find the direction and length of side BC.

  5. Length of BC

    BC=(9)2+(0)2=9BC = \sqrt{(9)^2+(0)^2} = 9

    Side BC is 9 units long.

  6. Check the corner is a right angle

    AB⃗⋅BC⃗=0\vec{AB} \cdot \vec{BC} = 0

    The two side vectors are perpendicular, confirming a rectangle.

  7. Check the fourth vertex

    D=A+BC⃗=(10,2)D = A + \vec{BC} = (10, 2)

    D completes the rectangle as given.

  8. Write the area formula

    Area=length×width\text{Area} = \text{length} \times \text{width}

    For a rectangle, multiply the two different side lengths.

  9. Substitute the side lengths

    Area=5×9\text{Area} = 5 \times 9

    Use AB and BC as the length and width.

  10. Work out the area

    Area=45\text{Area} = 45

    The area is 45 square units.

  11. Write the perimeter formula

    Perimeter=2(length+width)\text{Perimeter} = 2(\text{length}+\text{width})

    Add the four sides.

  12. Work out the perimeter

    P=2(5+9)=28P = 2(5+9) = 28

    The perimeter is 28 units.

  13. Sketch check

    plot A,B,C,D\text{plot } A,B,C,D

    The plotted points form the expected axis-aligned rectangle.

  14. Common mistake

    5+9≠Area5+9 \neq \text{Area}

    Area multiplies the sides; adding them gives half the perimeter, not the area.

  15. State the results

    Area=45,  P=28\text{Area} = 45,\; P = 28

    The rectangle has area 45 square units and perimeter 28 units.

Answer
Area = 45, perimeter = 28
Question 2
6 markschallenging
A triangle has vertices A=(0,0)A=(0, 0), B=(5,0)B=(5, 0) and C=(0,12)C=(0, 12). Work out its perimeter and classify the triangle.
Show worked solution

Worked solution

  1. List the vertices of the triangle

    A=(0,0),  B=(5,0),  C=(0,12)A=(0, 0),\; B=(5, 0),\; C=(0, 12)

    The perimeter is the sum of the three side lengths.

  2. Set up the distance formula

    d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

    Use it for each side in turn.

  3. Length of AB

    AB=(5)2+(0)2=25=5AB = \sqrt{(5)^2+(0)^2} = \sqrt{25} = 5

    Square the horizontal and vertical gaps, add, then square-root.

  4. Length of BC

    BC=(−5)2+(12)2=169=13BC = \sqrt{(-5)^2+(12)^2} = \sqrt{169} = 13

    Repeat for the side BC.

  5. Length of CA

    CA=(0)2+(−12)2=144=12CA = \sqrt{(0)^2+(-12)^2} = \sqrt{144} = 12

    And once more for CA.

  6. Add the three sides

    P=5+13+12P = 5 + 13 + 12

    The perimeter is the total of the side lengths.

  7. Simplify the perimeter

    P=30P = 30

    Add the whole numbers to get the perimeter.

  8. Classify the triangle

    all sides differ

    All three sides differ, so the triangle is scalene.

  9. Sketch check

    plot A,B,C\text{plot } A,B,C

    Plotting the points confirms the side lengths look right relative to the grid.

  10. State the perimeter

    P=30P = 30

    The perimeter of triangle ABC is 30 units.

  11. Test for a right angle

    25+144=16925+144 = 169

    The two shorter squared-sides add to the longest squared-side, so by the converse of Pythagoras there is a right angle.

  12. Recheck the longest side

    max⁡=169=13\max = \sqrt{169} = 13

    Confirm the largest side length before trusting the perimeter total.

  13. Common mistake

    count squares≠length\text{count squares} \neq \text{length}

    Counting grid squares along a slanted side undercounts; always use the distance formula.

  14. Reflect on the method

    d=(Δx)2+(Δy)2d = \sqrt{(\Delta x)^2+(\Delta y)^2}

    Each side length is a separate application of Pythagoras on the coordinate grid.

  15. Final statement

    P=30P = 30

    The exact perimeter is 30 units, and the triangle is scalene. It is also right-angled.

Answer
3030
Question 3
5 markschallenging
Show that the quadrilateral with vertices A=(1,0)A=(1, 0), B=(5,1)B=(5, 1), C=(6,4)C=(6, 4) and D=(2,3)D=(2, 3) is a parallelogram.
Show worked solution

Worked solution

  1. List the four vertices

    A=(1,0),  B=(5,1),  C=(6,4),  D=(2,3)A=(1, 0),\; B=(5, 1),\; C=(6, 4),\; D=(2, 3)

    We show ABCD is a parallelogram in two independent ways.

  2. Plan the proof

    diagonals bisect  &  opposite sides equal\text{diagonals bisect} \;\&\; \text{opposite sides equal}

    A quadrilateral is a parallelogram if its diagonals share a midpoint (or opposite sides are equal vectors).

  3. Midpoint of diagonal AC

    (1+62,0+42)\left(\frac{1+6}{2}, \frac{0+4}{2}\right)

    Average A and C.

  4. Evaluate midpoint of AC

    (3.5,2)(3.5, 2)

    The midpoint of AC is (3.5, 2).

  5. Midpoint of diagonal BD

    (5+22,1+32)\left(\frac{5+2}{2}, \frac{1+3}{2}\right)

    Average B and D.

  6. Evaluate midpoint of BD

    (3.5,2)(3.5, 2)

    The midpoint of BD is (3.5, 2).

  7. Compare the diagonal midpoints

    (3.5,2)=(3.5,2)(3.5, 2) = (3.5, 2)

    The diagonals share the same midpoint, so they bisect each other.

  8. First conclusion

    diagonals bisect⇒parallelogram\text{diagonals bisect} \Rightarrow \text{parallelogram}

    This alone is enough, but we confirm with the side vectors.

  9. Vector AB

    AB⃗=(4,1)\vec{AB} = (4, 1)

    Subtract A from B.

  10. Vector DC

    DC⃗=(4,1)\vec{DC} = (4, 1)

    Subtract D from C.

  11. Compare AB and DC

    AB⃗=DC⃗=(4,1)\vec{AB} = \vec{DC} = (4, 1)

    The vectors are equal, so AB and DC are equal and parallel.

  12. Second conclusion

    AB∥DC,  ∣AB∣=∣DC∣AB \parallel DC,\; |AB|=|DC|

    One pair of opposite sides is equal and parallel — another parallelogram test.

  13. Sketch check

    plot A,B,C,D\text{plot } A,B,C,D

    Plotting the points shows the expected slanted parallelogram.

  14. Common mistake

    AB⃗≠CD⃗\vec{AB} \neq \vec{CD}

    Compare AB with DC (not CD); direction matters when matching sides.

  15. State the conclusion

    ABCD is a parallelogramABCD \text{ is a parallelogram}

    Both tests pass (diagonals meet at (3.5, 2)), so ABCD is a parallelogram.

Answer
ABCD is a parallelogram
Question 4
6 markschallenging
The segment from A=(2,3)A=(2, 3) to B=(14,15)B=(14, 15) is divided into four equal parts. Find the three dividing points.
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Worked solution

  1. Write down the endpoints

    A=(2,3),  B=(14,15)A=(2, 3),\; B=(14, 15)

    We split AB into four equal parts, giving three dividing points.

  2. Find the vector AB

    AB⃗=(12,12)\vec{AB} = (12, 12)

    Subtract A from B to get the full step across the segment.

  3. Find one quarter of the vector

    14AB⃗=(3,3)\tfrac{1}{4}\vec{AB} = (3, 3)

    Each equal part moves this much.

  4. First point: A + a quarter

    P1=(2+3,3+3)P_1 = (2+3, 3+3)

    Add one quarter step to A.

  5. Evaluate the first point

    P1=(5,6)P_1 = (5, 6)

    The first dividing point is (5, 6).

  6. Second point: A + a half

    P2=(2+6,3+6)P_2 = (2+6, 3+6)

    This is the midpoint of AB.

  7. Evaluate the second point

    P2=(8,9)P_2 = (8, 9)

    The midpoint is (8, 9).

  8. Third point: A + three quarters

    P3=(2+9,3+9)P_3 = (2+9, 3+9)

    Add three quarter steps to A.

  9. Evaluate the third point

    P3=(11,12)P_3 = (11, 12)

    The third dividing point is (11, 12).

  10. Check the spacing to B

    P3+(3,3)=(14,15)P_3 + (3, 3) = (14, 15)

    One more quarter step from P3P_3 lands exactly on B, confirming equal spacing.

  11. Check P2P_2 is the midpoint

    mid(A,B)=(8,9)\text{mid}(A,B) = (8, 9)

    The middle dividing point matches the midpoint, as expected.

  12. Sketch check

    four equal gaps\text{four equal gaps}

    The three points cut AB into four segments of equal length.

  13. Common mistake

    three points, not four\text{three points, not four}

    Splitting into four parts needs three interior points, not four.

  14. Reflect on the method

    Pk=A+k4AB⃗P_k = A + \tfrac{k}{4}\vec{AB}

    Fractions of the vector AB give any point along the segment.

  15. State the three points

    P1=(5,6),  P2=(8,9),  P3=(11,12)P_1=(5, 6),\; P_2=(8, 9),\; P_3=(11, 12)

    The points dividing AB into four equal parts are (5, 6), (8, 9) and (11, 12).

Answer
(5,6),(8,9),(11,12)(5, 6), (8, 9), (11, 12)
Question 5
5 markschallenging
A=(0,1)A=(0, 1) and B=(6,13)B=(6, 13). Find the midpoint and the exact length of ABAB, and show the midpoint lies on the line y=2x+1y = 2x + 1.
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Worked solution

  1. Write down the endpoints

    A=(0,1),  B=(6,13)A=(0, 1),\; B=(6, 13)

    We will find the midpoint, the length, and check the midpoint lies on a given line.

  2. Midpoint formula

    M=(x1+x22,y1+y22)M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

    Average the coordinates for the midpoint.

  3. Substitute for the midpoint

    M=(0+62,1+132)M = \left(\frac{0+6}{2}, \frac{1+13}{2}\right)

    Put the coordinates into the formula.

  4. Evaluate the midpoint

    M=(3,7)M = (3, 7)

    The midpoint of AB is (3, 7).

  5. Distance formula

    d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

    Now find the exact length of AB.

  6. Horizontal change

    x2−x1=6x_2-x_1 = 6

    The x-coordinate changes by this amount.

  7. Vertical change

    y2−y1=12y_2-y_1 = 12

    The y-coordinate changes by this amount.

  8. Square and add

    (6)2+(12)2=36+144=180(6)^2+(12)^2 = 36+144 = 180

    Square both changes and add to get the squared length.

  9. Square root

    d=180d = \sqrt{180}

    Take the positive root.

  10. Simplify the length

    d=65d = 6\sqrt{5}

    The exact length of AB is 656\sqrt{5} units.

  11. Write the line to test

    y=2x+1y = 2x + 1

    Check whether the midpoint (3, 7) lies on this line.

  12. Substitute the midpoint's x

    y=2(3)+1y = 2(3) + 1

    Put the midpoint's x-coordinate into the line equation.

  13. Evaluate the right-hand side

    y=7y = 7

    The line gives y=7y = 7 at x=3x = 3.

  14. Compare with the midpoint's y

    7=77 = 7

    The value matches the midpoint's y-coordinate, so the midpoint lies on the line.

  15. State the results

    M=(3,7),  AB=65M = (3, 7),\; AB = 6\sqrt{5}

    The midpoint is (3, 7), the length is 656\sqrt{5}, and M lies on y=2x+1y = 2x + 1.

Answer
M=(3,7),length=65M = (3, 7), length = 6\sqrt{5}

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