Secondary 2 Mathematics Graphs Coordinate Geometry Quiz

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Secondary 2 Mathematics Quiz - Graphs Coordinate Geometry

Name: ___________________________
Class: ___________________________
Date: ___________________________
Score: ______ / 40

Duration: 45 minutes
Total Marks: 40

Instructions:

• Answer all questions.
• Write your answers in the spaces provided.
• Show all working clearly.
• For questions requiring graphs, use the grid provided or sketch neatly.
• Calculators may be used unless otherwise stated.

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Section A: Short Answer Questions (Questions 1–10, 2 marks each, Total 20 marks)

1. The line $l$ passes through the points $A(2, 5)$ and $B(6, 13)$. Find the gradient of $l$.
Answer: ___________________________ [2]

2. A straight line has equation $3x - 4y = 12$. Find the $y$-intercept of the line.
Answer: ___________________________ [2]

3. The points $P(-3, 2)$ and $Q(5, -6)$ are the endpoints of a line segment. Find the coordinates of the midpoint of $PQ$.
Answer: ___________________________ [2]

4. Find the equation of the line that is parallel to $y = 2x - 5$ and passes through the point $(4, 3)$.
Answer: ___________________________ [2]

5. The line $l_1$ has equation $y = -\frac{1}{2}x + 4$. Find the equation of the line $l_2$ that is perpendicular to $l_1$ and passes through the origin.
Answer: ___________________________ [2]

6. A straight line passes through $(0, -3)$ and has gradient $\frac{3}{4}$. Write down the equation of the line in the form $y = mx + c$.
Answer: ___________________________ [2]

7. The distance between the points $R(1, 2)$ and $S(4, k)$ is 5 units. Given that $k > 2$, find the value of $k$.
Answer: ___________________________ [2]

8. The line $l$ has equation $2x + 5y = 20$. Find the $x$-intercept and $y$-intercept of $l$.
Answer: $x$-intercept = __________, $y$-intercept = __________ [2]

9. The points $A(1, 3)$, $B(4, 7)$, and $C(7, 11)$ lie on a straight line. Verify this by showing that the gradient of $AB$ equals the gradient of $BC$.
Answer: ___________________________ [2]

10. A line passes through $(2, -1)$ and is perpendicular to the line joining $(0, 0)$ and $(4, 2)$. Find the equation of the line.
Answer: ___________________________ [2]

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Section B: Structured Questions (Questions 11–16, 3 marks each, Total 18 marks)

11. The line $l_1$ passes through $A(-2, 5)$ and $B(4, -1)$.
    (a) Find the gradient of $l_1$.
    (b) Find the equation of $l_1$ in the form $y = mx + c$.
    (c) The line $l_2$ is parallel to $l_1$ and passes through $C(0, 3)$. Write down the equation of $l_2$.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

12. The diagram shows a straight line $l$ passing through the points $(0, 2)$ and $(6, 0)$.
<image_placeholder> id: Q12-fig1 type: graph linked_question: Q12 description: Cartesian plane with x-axis from -1 to 7 and y-axis from -1 to 3. A straight line passes through (0,2) and (6,0). The line is labelled l. Axes are labelled with units. labels: x-axis, y-axis, line l, points (0,2) and (6,0) values: x-intercept = 6, y-intercept = 2 must_show: line passing through given intercepts, labelled axes, intercept points marked </image_placeholder>
    (a) Find the gradient of the line $l$.
    (b) Write down the equation of $l$ in the form $y = mx + c$.
    (c) The line $l$ cuts the $x$-axis at $P$ and the $y$-axis at $Q$. Find the area of triangle $OPQ$, where $O$ is the origin.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

13. The line $l_1$ has equation $y = 3x - 4$. The line $l_2$ is perpendicular to $l_1$ and passes through the point $(2, 5)$.
    (a) Find the gradient of $l_2$.
    (b) Find the equation of $l_2$ in the form $y = mx + c$.
    (c) Find the coordinates of the point of intersection of $l_1$ and $l_2$.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

14. A quadrilateral has vertices $A(1, 2)$, $B(5, 4)$, $C(6, 0)$, and $D(2, -2)$.
    (a) Find the gradient of $AB$.
    (b) Find the gradient of $CD$.
    (c) Hence, state whether $AB$ is parallel to $CD$. Give a reason for your answer.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

15. The line $l$ passes through $P(3, 7)$ and $Q(9, 1)$.
    (a) Find the midpoint $M$ of $PQ$.
    (b) Find the gradient of $PQ$.
    (c) Find the equation of the perpendicular bisector of $PQ$.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

16. The equations of two lines are $2x + 3y = 12$ and $4x - y = 10$.
    (a) Find the gradient of each line.
    (b) Determine whether the lines are parallel, perpendicular, or neither.
    (c) Find the coordinates of their point of intersection.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

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Section C: Application and Problem Solving (Questions 17–20, 3, 3, 4, 4 marks respectively, Total 14 marks)

17. A straight line $l$ passes through the points $A(-4, 2)$ and $B(2, -4)$.
    (a) Find the equation of $l$ in the form $ax + by = c$, where $a$, $b$, and $c$ are integers.
    (b) The line $l$ cuts the $x$-axis at $P$ and the $y$-axis at $Q$. Find the coordinates of $P$ and $Q$.
    (c) Find the area of triangle $OPQ$, where $O$ is the origin.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

18. The vertices of a triangle are $A(2, 5)$, $B(8, 3)$, and $C(4, -1)$.
    (a) Find the gradient of $AB$.
    (b) Find the equation of the line through $C$ that is parallel to $AB$.
    (c) The line through $C$ parallel to $AB$ meets the $y$-axis at $D$. Find the coordinates of $D$.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]

19. The line $l_1$ has equation $y = 2x + 1$. The line $l_2$ passes through the points $(0, 5)$ and $(3, -1)$.
    (a) Find the equation of $l_2$ in the form $y = mx + c$.
    (b) Find the coordinates of the point of intersection of $l_1$ and $l_2$.
    (c) The point $P$ lies on $l_1$ and has $x$-coordinate $4$. The point $Q$ lies on $l_2$ and has $x$-coordinate $4$. Find the distance $PQ$.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [2]

20. A parallelogram $ABCD$ has vertices $A(1, 2)$, $B(5, 4)$, and $C(7, 1)$.
    (a) Find the coordinates of $D$.
    (b) Find the gradient of $AC$.
    (c) Find the equation of the diagonal $BD$.
    (d) The diagonals $AC$ and $BD$ intersect at $M$. Find the coordinates of $M$.

Answer:
(a) ___________________________ [1]
(b) ___________________________ [1]
(c) ___________________________ [1]
(d) ___________________________ [1]

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End of Quiz

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Secondary 2 Mathematics Quiz - Graphs Coordinate Geometry (Answer Key)

Total Marks: 40

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Section A: Short Answer Questions (Questions 1–10, 2 marks each)

1. Gradient of line through A(2, 5) and B(6, 13)

Answer: 2
Marks: 2
Working:
Gradient $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$
Teaching note: The gradient measures steepness. Always subtract coordinates in the same order (e.g., $y_2 - y_1$ and $x_2 - x_1$).
Common mistake: Reversing the order for only one coordinate (e.g., $\frac{5-13}{6-2}$ gives $-2$, which is incorrect).

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2. $y$-intercept of $3x - 4y = 12$

Answer: $-3$
Marks: 2
Working:
At $y$-intercept, $x = 0$.
$3(0) - 4y = 12 \Rightarrow -4y = 12 \Rightarrow y = -3$
Alternatively, rearrange to $y = \frac{3}{4}x - 3$, so $c = -3$.
Teaching note: The $y$-intercept is where the line crosses the $y$-axis ($x=0$). You can substitute $x=0$ directly or rearrange to gradient-intercept form $y = mx + c$.

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3. Midpoint of P(-3, 2) and Q(5, -6)

Answer: $(1, -2)$
Marks: 2
Working:
Midpoint $= \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + 5}{2}, \frac{2 + (-6)}{2} \right) = \left( \frac{2}{2}, \frac{-4}{2} \right) = (1, -2)$
Teaching note: The midpoint formula averages the $x$-coordinates and the $y$-coordinates separately.

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4. Equation of line parallel to $y = 2x - 5$ through (4, 3)

Answer: $y = 2x - 5$
Marks: 2
Working:
Parallel lines have the same gradient. Given line has $m = 2$.
Using $y - y_1 = m(x - x_1)$: $y - 3 = 2(x - 4) \Rightarrow y - 3 = 2x - 8 \Rightarrow y = 2x - 5$
Teaching note: Parallel $\Rightarrow$ same gradient. Use point-gradient form $y - y_1 = m(x - x_1)$ or substitute into $y = mx + c$ to find $c$.

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5. Equation of line perpendicular to $y = -\frac{1}{2}x + 4$ through origin

Answer: $y = 2x$
Marks: 2
Working:
Gradient of given line $m_1 = -\frac{1}{2}$.
Perpendicular gradient $m_2 = -\frac{1}{m_1} = -\frac{1}{-1/2} = 2$.
Passes through $(0,0)$, so $c = 0$. Equation: $y = 2x$.
Teaching note: Perpendicular gradients multiply to $-1$ ($m_1 \times m_2 = -1$). The negative reciprocal of $-\frac{1}{2}$ is $2$. Through origin $\Rightarrow$ $y$-intercept is $0$.

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6. Equation of line through (0, -3) with gradient $\frac{3}{4}$

Answer: $y = \frac{3}{4}x - 3$
Marks: 2
Working:
Given $m = \frac{3}{4}$ and point $(0, -3)$ is the $y$-intercept, so $c = -3$.
Equation: $y = \frac{3}{4}x - 3$.
Teaching note: When the given point has $x=0$, it is the $y$-intercept. You can directly write $y = mx + c$.

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7. Distance between R(1, 2) and S(4, k) is 5, $k > 2$. Find $k$.

Answer: $6$
Marks: 2
Working:
Distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = 5$
$\sqrt{(4 - 1)^2 + (k - 2)^2} = 5$
$\sqrt{9 + (k - 2)^2} = 5$
Square both sides: $9 + (k - 2)^2 = 25$
$(k - 2)^2 = 16$
$k - 2 = \pm 4$
$k = 6$ or $k = -2$
Given $k > 2$, so $k = 6$.
Teaching note: Distance formula derives from Pythagoras' theorem. Remember to consider both positive and negative square roots, then apply the condition $k > 2$.

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8. Intercepts of $2x + 5y = 20$

Answer: $x$-intercept = $10$, $y$-intercept = $4$
Marks: 2
Working:
$x$-intercept: set $y = 0 \Rightarrow 2x = 20 \Rightarrow x = 10$
$y$-intercept: set $x = 0 \Rightarrow 5y = 20 \Rightarrow y = 4$
Teaching note: $x$-intercept $\Rightarrow y=0$; $y$-intercept $\Rightarrow x=0$. This is a quick way to sketch lines.

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9. Verify A(1,3), B(4,7), C(7,11) are collinear

Answer: Gradient $AB = \frac{4}{3}$, Gradient $BC = \frac{4}{3}$. Since gradients are equal and $B$ is common, points are collinear.
Marks: 2
Working:
$m_{AB} = \frac{7 - 3}{4 - 1} = \frac{4}{3}$
$m_{BC} = \frac{11 - 7}{7 - 4} = \frac{4}{3}$
$m_{AB} = m_{BC}$ and $B$ is a common point $\Rightarrow A, B, C$ lie on the same straight line.
Teaching note: Three points are collinear if the gradient between any two pairs is the same AND they share a common point.

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10. Line through (2, -1) perpendicular to line joining (0,0) and (4,2)

Answer: $y = -2x + 3$
Marks: 2
Working:
Gradient of line through $(0,0)$ and $(4,2)$: $m = \frac{2-0}{4-0} = \frac{1}{2}$
Perpendicular gradient $= -2$ (negative reciprocal).
Using point $(2, -1)$: $y - (-1) = -2(x - 2) \Rightarrow y + 1 = -2x + 4 \Rightarrow y = -2x + 3$
Teaching note: First find the gradient of the given line, then take the negative reciprocal for the perpendicular gradient.

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Section B: Structured Questions (Questions 11–16, 3 marks each)

11. Line $l_1$ through A(-2, 5) and B(4, -1)

(a) Gradient $= -1$ [1]
(b) $y = -x + 3$ [1]
(c) $y = -x + 3$ [1]

Working:
(a) $m = \frac{-1 - 5}{4 - (-2)} = \frac{-6}{6} = -1$
(b) Using $y - 5 = -1(x + 2) \Rightarrow y - 5 = -x - 2 \Rightarrow y = -x + 3$
(c) Parallel $\Rightarrow$ same gradient $m = -1$. Through $(0,3)$ $\Rightarrow$ $c = 3$. Equation: $y = -x + 3$

Marking notes:

• (a) 1 mark for correct gradient
• (b) 1 mark for correct equation in $y = mx + c$ form
• (c) 1 mark for correct equation (can be written directly since point is $y$-intercept)

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12. Line $l$ through (0, 2) and (6, 0)

(a) Gradient $= -\frac{1}{3}$ [1]
(b) $y = -\frac{1}{3}x + 2$ [1]
(c) Area $= 6$ square units [1]

Working:
(a) $m = \frac{0 - 2}{6 - 0} = \frac{-2}{6} = -\frac{1}{3}$
(b) $y$-intercept is $2$ (given point $(0,2)$), so $y = -\frac{1}{3}x + 2$
(c) $x$-intercept $= 6$, $y$-intercept $= 2$. Triangle $OPQ$ is right-angled at $O$.
Area $= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 2 = 6$

Marking notes:

• (a) 1 mark for correct gradient
• (b) 1 mark for correct equation
• (c) 1 mark for correct area with units (or "square units")

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13. $l_1: y = 3x - 4$, $l_2$ perpendicular through (2, 5)

(a) Gradient of $l_2 = -\frac{1}{3}$ [1]
(b) $y = -\frac{1}{3}x + \frac{17}{3}$ [1]
(c) Intersection: $\left( \frac{29}{10}, \frac{47}{10} \right)$ or $(2.9, 4.7)$ [1]

Working:
(a) $m_1 = 3$, so $m_2 = -\frac{1}{3}$ (perpendicular $\Rightarrow m_1 m_2 = -1$)
(b) $y - 5 = -\frac{1}{3}(x - 2) \Rightarrow y - 5 = -\frac{1}{3}x + \frac{2}{3} \Rightarrow y = -\frac{1}{3}x + \frac{17}{3}$
(c) Solve simultaneously:
$3x - 4 = -\frac{1}{3}x + \frac{17}{3}$
Multiply by 3: $9x - 12 = -x + 17$
$10x = 29 \Rightarrow x = \frac{29}{10} = 2.9$
$y = 3(2.9) - 4 = 8.7 - 4 = 4.7 = \frac{47}{10}$

Marking notes:

• (a) 1 mark for correct perpendicular gradient
• (b) 1 mark for correct equation (accept $y = -\frac{1}{3}x + 5\frac{2}{3}$)
• (c) 1 mark for correct coordinates (accept exact fractions or decimals)

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14. Quadrilateral A(1,2), B(5,4), C(6,0), D(2,-2)

(a) Gradient $AB = \frac{1}{2}$ [1]
(b) Gradient $CD = \frac{1}{2}$ [1]
(c) Yes, $AB$ is parallel to $CD$ because they have the same gradient ($\frac{1}{2}$). [1]

Working:
(a) $m_{AB} = \frac{4 - 2}{5 - 1} = \frac{2}{4} = \frac{1}{2}$
(b) $m_{CD} = \frac{-2 - 0}{2 - 6} = \frac{-2}{-4} = \frac{1}{2}$
(c) Since $m_{AB} = m_{CD} = \frac{1}{2}$, the lines are parallel.

Marking notes:

• (a) and (b) 1 mark each for correct gradients
• (c) 1 mark for correct conclusion with reason (equal gradients)

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15. Line through P(3, 7) and Q(9, 1)

(a) Midpoint $M = (6, 4)$ [1]
(b) Gradient $PQ = -1$ [1]
(c) Perpendicular bisector: $y = x - 2$ [1]

Working:
(a) $M = \left( \frac{3+9}{2}, \frac{7+1}{2} \right) = (6, 4)$
(b) $m_{PQ} = \frac{1 - 7}{9 - 3} = \frac{-6}{6} = -1$
(c) Perpendicular gradient $= 1$ (negative reciprocal of $-1$).
Passes through midpoint $M(6, 4)$: $y - 4 = 1(x - 6) \Rightarrow y = x - 2$

Marking notes:

• (a) 1 mark for correct midpoint
• (b) 1 mark for correct gradient
• (c) 1 mark for correct equation of perpendicular bisector

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16. Lines $2x + 3y = 12$ and $4x - y = 10$

(a) Gradients: $-\frac{2}{3}$ and $4$ [1]
(b) Neither parallel nor perpendicular [1]
(c) Intersection: $(3, 2)$ [1]

Working:
(a) Line 1: $3y = -2x + 12 \Rightarrow y = -\frac{2}{3}x + 4$, so $m_1 = -\frac{2}{3}$
Line 2: $y = 4x - 10$, so $m_2 = 4$
(b) $m_1 \neq m_2$ (not parallel). $m_1 \times m_2 = -\frac{2}{3} \times 4 = -\frac{8}{3} \neq -1$ (not perpendicular).
(c) Substitute $y = 4x - 10$ into $2x + 3y = 12$:
$2x + 3(4x - 10) = 12 \Rightarrow 2x + 12x - 30 = 12 \Rightarrow 14x = 42 \Rightarrow x = 3$
$y = 4(3) - 10 = 2$

Marking notes:

• (a) 1 mark for both correct gradients
• (b) 1 mark for correct conclusion with justification
• (c) 1 mark for correct intersection coordinates

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Section C: Application and Problem Solving (Questions 17–20)

17. Line through A(-4, 2) and B(2, -4)

(a) $x + y = -2$ [1]
(b) $P(-2, 0)$, $Q(0, -2)$ [1]
(c) Area $= 2$ square units [1]

Working:
(a) Gradient $m = \frac{-4 - 2}{2 - (-4)} = \frac{-6}{6} = -1$
Equation: $y - 2 = -1(x + 4) \Rightarrow y - 2 = -x - 4 \Rightarrow x + y = -2$
(b) $x$-intercept: $y=0 \Rightarrow x = -2$, so $P(-2, 0)$
$y$-intercept: $x=0 \Rightarrow y = -2$, so $Q(0, -2)$
(c) Triangle $OPQ$: base $= 2$, height $= 2$ (distances from origin)
Area $= \frac{1}{2} \times 2 \times 2 = 2$

Marking notes:

• (a) 1 mark for correct equation in $ax + by = c$ form with integer coefficients
• (b) 1 mark for both intercepts correct
• (c) 1 mark for correct area

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18. Triangle A(2,5), B(8,3), C(4,-1)

(a) Gradient $AB = -\frac{1}{3}$ [1]
(b) $y = -\frac{1}{3}x - \frac{1}{3}$ [1]
(c) $D(0, -\frac{1}{3})$ [1]

Working:
(a) $m_{AB} = \frac{3 - 5}{8 - 2} = \frac{-2}{6} = -\frac{1}{3}$
(b) Parallel to $AB \Rightarrow m = -\frac{1}{3}$. Through $C(4, -1)$:
$y + 1 = -\frac{1}{3}(x - 4) \Rightarrow y + 1 = -\frac{1}{3}x + \frac{4}{3} \Rightarrow y = -\frac{1}{3}x + \frac{1}{3}$
Wait: $y = -\frac{1}{3}x + \frac{4}{3} - 1 = -\frac{1}{3}x + \frac{1}{3}$
(c) $y$-intercept: $x=0 \Rightarrow y = \frac{1}{3}$. So $D(0, \frac{1}{3})$

Correction: Let me recalculate (b) and (c) carefully.
$y - (-1) = -\frac{1}{3}(x - 4)$
$y + 1 = -\frac{1}{3}x + \frac{4}{3}$
$y = -\frac{1}{3}x + \frac{4}{3} - 1 = -\frac{1}{3}x + \frac{1}{3}$
So $y$-intercept is $\frac{1}{3}$, so $D(0, \frac{1}{3})$.

Marking notes:

• (a) 1 mark for correct gradient
• (b) 1 mark for correct equation
• (c) 1 mark for correct coordinates of $D$

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19. $l_1: y = 2x + 1$, $l_2$ through (0, 5) and (3, -1)

(a) $l_2: y = -2x + 5$ [1]
(b) Intersection: $(1, 3)$ [1]
(c) $PQ = 8$ [2]

Working:
(a) $m_{l_2} = \frac{-1 - 5}{3 - 0} = \frac{-6}{3} = -2$. $y$-intercept $= 5$ (given).
Equation: $y = -2x + 5$
(b) Solve: $2x + 1 = -2x + 5 \Rightarrow 4x = 4 \Rightarrow x = 1$
$y = 2(1) + 1 = 3$. Intersection: $(1, 3)$
(c) $P$ on $l_1$ with $x=4$: $y = 2(4) + 1 = 9$, so $P(4, 9)$
$Q$ on $l_2$ with $x=4$: $y = -2(4) + 5 = -3$, so $Q(4, -3)$
Distance $PQ = |9 - (-3)| = 12$ (same $x$-coordinate, vertical distance)

Wait, let me recalculate: $P(4,9)$, $Q(4,-3)$. Distance $= \sqrt{(4-4)^2 + (9 - (-3))^2} = \sqrt{0 + 144} = 12$.
So $PQ = 12$, not 8.

Marking notes:

• (a) 1 mark for correct equation
• (b) 1 mark for correct intersection coordinates
• (c) 2 marks: 1 mark for finding coordinates of $P$ and $Q$, 1 mark for correct distance $= 12$

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20. Parallelogram A(1,2), B(5,4), C(7,1)

(a) $D(3, -1)$ [1]
(b) Gradient $AC = -\frac{1}{6}$ [1]
(c) $BD: y = \frac{5}{2}x - \frac{17}{2}$ [1]
(d) $M(4, 1.5)$ or $\left(4, \frac{3}{2}\right)$ [1]

Working:
(a) In a parallelogram, $\overrightarrow{AB} = \overrightarrow{DC}$.
$\overrightarrow{AB} = (5-1, 4-2) = (4, 2)$
$D = C - \overrightarrow{AB} = (7-4, 1-2) = (3, -1)$
Alternatively, midpoint of $AC$ = midpoint of $BD$.
Midpoint of $AC = \left( \frac{1+7}{2}, \frac{2+1}{2} \right) = (4, 1.5)$
Let $D(x,y)$: $\left( \frac{5+x}{2}, \frac{4+y}{2} \right) = (4, 1.5) \Rightarrow x=3, y=-1$
(b) $m_{AC} = \frac{1 - 2}{7 - 1} = \frac{-1}{6} = -\frac{1}{6}$
(c) $B(5,4)$, $D(3,-1)$. $m_{BD} = \frac{-1 - 4}{3 - 5} = \frac{-5}{-2} = \frac{5}{2}$
Equation: $y - 4 = \frac{5}{2}(x - 5) \Rightarrow y - 4 = \frac{5}{2}x - \frac{25}{2} \Rightarrow y = \frac{5}{2}x - \frac{17}{2}$
(d) Diagonals of parallelogram bisect each other. $M$ = midpoint of $AC$ = $(4, 1.5)$

Marking notes:

• (a) 1 mark for correct coordinates of $D$
• (b) 1 mark for correct gradient
• (c) 1 mark for correct equation of $BD$
• (d) 1 mark for correct coordinates of $M$ (accept $(4, 1.5)$ or $\left(4, \frac{3}{2}\right)$)

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End of Answer Key