Grade 10 Maths · Analytical Geometry · Show that points are collinear

Show that points are collinear Grade 10

Show that three points lie on one straight line by showing the gradients between them are equal, or use equal gradients to find an unknown coordinate.

questions use it, all grades · 6 past paper, 14 practice: 20 questions use it, all grades · 6 past paper, 14 practice
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Analytical Geometry Collinear Points Grade 10 MathsMiss Martins Maths and Science

3 Gr 10 Mathematics - Analytical Geometry - Collinear PointsJuniorTukkie at the University of Pretoria

Analytical Geometry Grade 10 | Collinear Points Explained Gridwell Maths Tutoring

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Question 4.1 · November 2016 · Paper 2 · National (DBE) · 3 marks

In the diagram, R and A are the

x

- and

y

-intercepts respectively of the straight line AR. The equation of AR is

y = - \frac{1}{2} x + 4

. Another straight line cuts the

y

-axis at

P (0; 2)

and passes through the points

M (k; 0)

and

N (3; 4)

α

and

β

are the angles of inclination of the lines MN and AR respectively. $Diagram for this question$

4.1 Given that M, P and N are collinear points, calculate the value of

k

.(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More questions: show that points are collinear

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3. Question 3.5 · March 2016 · Provincial · 3 marks
3.5 If the following points $B (x; - 3)$ , $E (- 4; 5)$ and $F (2; 2)$ are collinear, calculate the value of x.(3)
Show the memo (3 ticks)Memo
1. ✓
  Equating gradients (M) $m_{BE} = m_{EF}$ : $\frac{5 - ( - 3 )}{- 4 - x} = \frac{2 - 5}{2 - ( - 4 )}$
  1 mark · Show that points are collinear
2. ✓
  Correct gradients $\frac{8}{- 4 - x} = - \frac{1}{2}$
  1 mark · Calculate gradient between two points
3. ✓
  Answer $16 = - 1 (- 4 - x) = 4 + x$ , so $x = 12$
  1 mark · Show that points are collinear
4. Question 4.2.3 · March 2018 · Provincial · 4 marks
4.2 In the diagram $P (3; 7), Q (- 3; 4)$ and $R (1; - 6)$ are vertices of a triangle.
4.2.3 N is the point $(3; - 11)$ . Are the points N, R and Q collinear? Justify your answer by means of calculations.(4)
Show the memo (4 ticks)Memo
1. ✓
  Gradient of NR $m_{NR} = \frac{- 6 - ( - 11 )}{1 - 3} = \frac{5}{- 2} = - 2, 5$
  2 marks · Calculate gradient between two points
2. ✓
  Compare gradients $m_{NR} = - 2, 5$ and $m_{RQ} = - 2, 5$ (calculated in 4.2.2)
  1 mark · Show that points are collinear
3. ✓
  ConclusionSince gradients are equal and R is a common point, N, R and Q are collinear.
  1 mark · Show that points are collinear
5. Question 3.1 · November 2023 · Paper 2 · Provincial · 5 marks
3.1 Collinear points are points that lie on the same straight line. If A $(- 8; 0)$ , B $(x - 5; - 8)$ and C $(x; - 14)$ are collinear, then calculate the value of $x$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  Gradient of AB $m_{A B} = m_{A C}$
  $\frac{- 8 - 0}{x - 5 + 8}$
  1 mark · Calculate gradient between two points
2. ✓
  Gradient of AC $= \frac{- 14 - 0}{x + 8}$
  $\frac{- 8}{x + 3} = \frac{- 14}{x + 8}$
  1 mark · Calculate gradient between two points
3. ✓
  Cross-multiplying (left side) $- 8 x - 64$
  1 mark · Clear fractions in an equation
4. ✓
  Cross-multiplying (right side) $- 8 x - 64 = - 14 x - 42$
  1 mark · Clear fractions in an equation
5. ✓
  Answer $6 x = 22$
  $x = \frac{11}{3}$
  1 mark · Show that points are collinear
6. Practice · Siyavula Grade 10 Maths · Ex 8.6 · 2 marks
On the Cartesian plane, the three points $P (- 3; 4)$ , $Q (7; - 1)$ and $R (3; b)$ are collinear.
(c) Find the value of $b$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Equal gradients $m_{PR} = m_{PQ}$
  $\frac{b - 4}{3 - ( - 3 )} = - \frac{1}{2}$
  1 mark · Show that points are collinear
2. ✓
  Answer $\frac{b - 4}{6} = - \frac{1}{2}$
  $b - 4 = - 3$
  $b = 1$
  1 mark · Show that points are collinear
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue

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