Grade 10 Maths · Analytical Geometry

Calculate the mid-point of a segment

Find the point halfway between two points by averaging their coordinates, for example (2; 4) and (6; 8) have mid-point (4; 6).

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Analytical Geometry Grade 10: MidpointKevinmathscience

Analytical Geometry Midpoint formula: Grade 10 MathsMiss Martins Maths and Science

Grade 11 analytical geometry: 4th Co-ordinate of parallelogramKevinmathscience

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Question 1.2 · June 2024 · Paper 2 · Provincial · 3 marks

In the figure below,

A (- 5; 2)

and

B (4; - 2)

are points on a line segment AB. Diagram for this question

1.2 Determine the coordinates of the midpoint of AB.(3)

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More past paper questions: calculate the mid-point of a segment

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3. Question 1.2 · June 2024 · Paper 2 · Provincial · 2 marks
$PQRS$ is a quadrilateral with vertices $P (- 5; 0)$ , $Q (0; t)$ , $R (3; - 2)$ and $S (- 2; - 5)$ .
$P$ is on the x-axis and $Q$ is on the y-axis.
1.2 Calculate the coordinates $M$ , the midpoint of $PR$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  x-value of M $M (\frac{- 5 + 3}{2}; \frac{0 - 2}{2})$ $x_{M} = - 1$
  1 mark · Calculate the mid-point of a segment
2. ✓
  y-value of M $y_{M} = - 1$ , so $M (- 1; - 1)$
  1 mark · Calculate the mid-point of a segment
4. Question 1.2 · June 2024 · Paper 2 · Provincial · 2 marks
In the diagram below, $A (- 2; 2)$ , $B (- 4; - 6)$ , $C (4; - 2)$ and $E (6; y)$ are the vertices of a quadrilateral with $A E ∣∣ BC$ . $D$ lies on $BC$ such that $A D ⊥ BC$ and $A C$ is drawn. $M (x; 2)$ is a point on line $EC$ .
1.2 If $M$ is the midpoint of $EC$ , determine the value of $x$ and $y$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Use midpoint formula for y $\frac{y + ( - 2 )}{2} = 2 ⟹ y - 2 = 4 ⟹ y = 6$
  1 mark · Calculate the mid-point of a segment
2. ✓
  Use midpoint formula for x $\frac{6 + 4}{2} = x ⟹ x = 5$
  1 mark · Calculate the mid-point of a segment
5. Question 3.3.3 · March 2016 · Provincial · 2 marks
3.3 In the diagram below, $A (- 1; 2)$ , $B (x; y)$ and $C (1; - 1)$ are the vertices of a triangle. D is the x-intercept and the midpoint of AB. $B \hat{A} C$ is $78, 6 9^{\circ}$ .
3.3.3 Determine the coordinates of B.(2)
Show the memo (2 ticks)Memo
1. ✓
  x-coordinate of B $\frac{- 1 + x _{B}}{2} = - 3 \Rightarrow x_{B} = - 5$
  1 mark · Calculate the mid-point of a segment
2. ✓
  y-coordinate of B $\frac{2 + y _{B}}{2} = 0 \Rightarrow y_{B} = - 2$
  $B (- 5; - 2)$
  1 mark · Calculate the mid-point of a segment
6. Question 4.5 · May/June 2025 · Paper 2 · National (DBE) · 3 marks
In the diagram, M is the centre of the circle having equation $(x - 3)^{2} + y^{2} = 25$ . $E (- 1; 3)$ and T are points on the circle.

EC is a tangent to the circle at E and cuts the x-axis at D. $E D = \frac{15}{4}$ units. MT is produced to meet the tangent at $C (- 7; p)$ .
4.5 Calculate the coordinates of S if SEMC is a parallelogram and $x_{S} < 0$ .(3)
Show the memo (1 tick)Memo
1. ✓
  METHOD 1: Using translation propertiesIn parallelogram SEMC, vector $ME = v ec t or CS$ .
  Translation from M(3; 0) to E(-1; 3):
  $x : 3 \to - 1$ (Subtract 4)
  $y : 0 \to 3$ (Add 3)
  
  Apply the same translation to C(-7; -5) to find S:
  $x_{S} = - 7 - 4 = - 11$
  $y_{S} = - 5 + 3 = - 2$
  $∴ S (- 11; - 2)$
  3 marks · Reflect, translate or rotate a point

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