Chapter 1 · III · Coordinate Geometry

The Use of Coordinates Worksheet

17 questions Cartesian coordinates Distance and midpoint Floor plans Collinearity and right triangles

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Foundation

Cartesian axes, quadrants and relative position · 1 practice mark

Question 1

The point (−3,4) lies in:

  1. Quadrant I
  2. Quadrant II
  3. Quadrant III
  4. Quadrant IV
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(b) Quadrant II

  1. Its x-coordinate is negative and y-coordinate positive.
  2. This sign combination identifies Quadrant II.

Distance formula · 1 practice mark

Question 2

The distance between (1,2) and (4,6) is:

  1. 3
  2. 4
  3. 5
  4. 7
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(c) 5

  1. The horizontal and vertical changes are 3 and 4.
  2. Distance = √(3² + 4²) = 5.

Cartesian axes, quadrants and relative position · 1 practice mark

Question 3

A point on the y-axis must have:

  1. y = 0
  2. x = 0
  3. x = y
  4. x = 1
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(b) x = 0

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

Midpoint formula · 1 practice mark

Question 4

The midpoint of (−5,−2) and (1,4) is:

  1. (−2,1)
  2. (−4,2)
  3. (2,−1)
  4. (−3,3)
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(a) (−2,1)

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

Axes and quadrants · 1 practice mark

Question 5

Where is the point (0,−5)?

  1. Quadrant III
  2. Quadrant IV
  3. Negative y-axis
  4. Positive x-axis
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(c) Negative y-axis

  1. Its horizontal coordinate is zero, so it is on the y-axis.
  2. Its vertical coordinate is negative, placing it below the origin.

Practice

Coordinate geometry as a link between algebra and geometry · Cartesian axes, quadrants and relative position · 3 practice marks

Question 6

Explain how assigning coordinates to a geometric point connects geometry with algebra. On a grid, mark A(2,1) and B(−1,3), and describe B relative to A.

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B is 3 units left and 2 units above A; coordinates allow geometric properties to be expressed algebraically.

  1. Two perpendicular axes and a common origin fix the reference system.
  2. An ordered pair records signed horizontal and vertical displacements; equations can then describe sets of geometric points.
  3. B − A = (−1 − 2, 3 − 1) = (−3,2). Plot using these displacements.

Midpoint formula · 3 practice marks

Question 7

Find the midpoint of A(−2,5) and B(6,−1). Verify it by comparing the displacements from each endpoint.

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M = (2,2).

  1. Average x-coordinates: (−2 + 6)/2 = 2.
  2. Average y-coordinates: (5 − 1)/2 = 2.
  3. M − A = (4,−3) and B − M = (4,−3), so M is equally placed along the segment.

Grid floor plans and modelling · Distance formula · 3 practice marks

Question 8

A rectangular classroom has grid vertices (0,0), (8,0), (8,6), (0,6), with 1 grid unit = 1 metre. Mark the centre and find the straight-line distance from (0,0) to the opposite corner.

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Centre (4,3); diagonal 10 m.

  1. Draw a rectangle 8 units wide and 6 units high using the given coordinates.
  2. The centre is the midpoint of opposite corners: ((0 + 8)/2, (0 + 6)/2) = (4,3).
  3. The diagonal length is √(8² + 6²) = √100 = 10 m.
Eight by six classroom floor plan with its centre and ten-unit diagonal(0,0)(8,6)(4,3)8 m6 m

Testing collinearity · Distance formula · 3 practice marks

Question 9

Use distances to decide whether A(0,0), B(2,2) and C(5,5) are collinear.

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They are collinear, with B between A and C.

  1. AB = √8 = 2√2; BC = √18 = 3√2.
  2. AC = √50 = 5√2.
  3. AB + BC = AC, so equality in the triangle inequality places B on segment AC.

Testing right-angled triangles · Distance formula · 3 practice marks

Question 10

Check whether A(0,0), B(4,0), C(0,3) form a right-angled triangle. Identify the right-angle vertex.

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The triangle is right-angled at A.

  1. AB² = 16 and AC² = 9.
  2. BC² = (4 − 0)² + (0 − 3)² = 25.
  3. AB² + AC² = BC²; by the converse of Pythagoras the angle opposite BC, at A, is 90°.

Midpoint formula · Cartesian axes, quadrants and relative position · 3 practice marks

Question 11

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

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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.

Coordinate geometry as a link between algebra and geometry · Cartesian axes, quadrants and relative position · 3 practice marks

Question 12

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.

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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.

Distance and midpoint · 3 practice marks

Question 13

Find the distance and midpoint of P(2,−3) and Q(8,5). Explain the different roles of differences and averages.

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PQ = 10 units; midpoint (5,1).

  1. The changes are 8 − 2 = 6 and 5 − (−3) = 8.
  2. Pythagoras gives PQ = √(6² + 8²) = 10.
  3. Average the respective coordinates: ((2 + 8)/2, (−3 + 5)/2) = (5,1). Differences measure displacement; averages locate the halfway point.

Challenge

Distance formula · 4 practice marks

Question 14

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).

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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.

Testing collinearity · Testing right-angled triangles · 4 practice marks

Question 15

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).

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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.

Grid floor plans and modelling · Distance formula · Midpoint formula · 4 practice marks

Question 16

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.

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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.

Exact collinearity · 4 practice marks

Question 17

P(1,3), Q(4,9) and R(t,2t + 1) are given. Prove R lies on the line PQ for every real t. When is R strictly between P and Q?

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R lies on y = 2x + 1; it is strictly between P and Q when 1 < t < 4.

  1. P and Q both satisfy y = 2x + 1.
  2. Their x-coordinates differ, so they determine this one line.
  3. R also satisfies the equation by construction.
  4. Along this non-vertical line the x-coordinate orders the points. Endpoints correspond to t = 1 and t = 4.