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Unit III: Coordinate Geometry — Test 2

Reasoning, proof and modelling · 7 questions · 20 practice marks · 40 minutes.
MCQs are scored automatically; written answers use step-wise self/teacher review.

Section A
Question 1 (1 Mark)
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Full question paper and step-wise model answers

Practice marks are not official examination weightage. Written answers require self- or teacher assessment.

Question 1 · 1 mark

A point on the y-axis must have:

  1. y = 0
  2. x = 0
  3. x = y
  4. x = 1
Model answer and reasoning

(b) x = 0

  1. The y-axis has no horizontal displacement from the origin.
  2. Every point on it therefore has x-coordinate 0.

Question 2 · 1 mark

The midpoint of (−5,−2) and (1,4) is:

  1. (−2,1)
  2. (−4,2)
  3. (2,−1)
  4. (−3,3)
Model answer and reasoning

(a) (−2,1)

  1. Average the respective coordinates, not cross-coordinates.
  2. ((−5 + 1)/2, (−2 + 4)/2) = (−2,1).

Question 3 · 4 marks

Derive the distance formula for A(x₁,y₁) and B(x₂,y₂) using a right triangle. Apply it to A(−3,−2) and B(5,4).

Model answer and reasoning

AB = √((x₂ − x₁)² + (y₂ − y₁)²); the example distance is 10.

  1. Draw horizontal and vertical segments between the two points to form a right triangle.
  2. Their lengths are |x₂ − x₁| and |y₂ − y₁|.
  3. Pythagoras gives the squared distance as the sum of the two squared coordinate differences.
  4. For the example, the differences are 8 and 6; √(64 + 36) = 10.

Question 4 · 3 marks

M(3,−1) is the midpoint of A(−2,4) and B. Find B and state its quadrant.

Model answer and reasoning

B = (8,−6), in Quadrant IV.

  1. (−2 + xB)/2 = 3 gives xB = 8.
  2. (4 + yB)/2 = −1 gives yB = −6.
  3. A positive x and negative y place B in Quadrant IV.

Question 5 · 4 marks

Describe a systematic algorithm to check if three distinct coordinate points form a triangle, and then whether that triangle is right-angled. Test A(1,1), B(4,1), C(4,5).

Model answer and reasoning

The points form a triangle right-angled at B.

  1. First test collinearity, for example whether (xB − xA)(yC − yA) = (yB − yA)(xC − xA). If equal, there is no non-degenerate triangle.
  2. For the given points the products are 12 and 0, so they are not collinear.
  3. Compute the three squared distances: AB² = 9, BC² = 16, AC² = 25.
  4. Sort these values. The two smaller sum to the largest, so the angle at B is right.

Question 6 · 4 marks

On a floor-plan grid, a straight corridor joins P(−1,2) to Q(7,8), with each unit representing 0.5 m. Find its actual length and the coordinates of a camera placed halfway along it.

Model answer and reasoning

Length 5 m; camera at (3,5).

  1. The grid displacement is (8,6), so the grid length is √100 = 10.
  2. Multiply by the scale 0.5 m per unit to get 5 m.
  3. The halfway coordinates are ((−1 + 7)/2, (2 + 8)/2) = (3,5).
  4. Coordinates remain in grid units; do not multiply them by the physical length scale unless explicitly converting the coordinate system.

Question 7 · 3 marks

Explain why an ordered pair must be ordered. Plot P(2,5) and Q(5,2) and explain how coordinate geometry lets us compare their positions without relying on an approximate picture.

Model answer and reasoning

The first coordinate is horizontal and the second vertical; P and Q are different points.

  1. An ordered pair assigns different roles to its first and second entries.
  2. P is 2 units right and 5 up; Q is 5 right and 2 up.
  3. Subtracting coordinates gives Q − P = (3,−3). Algebra makes this comparison exact and independent of drawing accuracy.