Verify collinearity or parallel/perpendicular relationship

Show or verify that points are collinear, or that two lines are parallel or perpendicular, typically by computing and comparing gradients.

7 questions · Moderate -0.7

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OCR MEI C1 Q4
13 marks Standard +0.3
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01569a16-66ba-422e-a74d-6f9430dd245b-2_592_782_322_730} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).
  1. Verify that the lines AB and DC are parallel.
  2. Prove that the trapezium is not isosceles.
  3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
  4. Show that neither diagonal of the trapezium bisects the other.
OCR MEI C1 2010 January Q10
13 marks Moderate -0.3
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-3_590_780_347_680} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).
  1. Verify that the lines AB and DC are parallel.
  2. Prove that the trapezium is not isosceles.
  3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
  4. Show that neither diagonal of the trapezium bisects the other.
Edexcel AS Paper 1 2018 June Q4
4 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)
The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.
AQA AS Paper 1 2018 June Q1
1 marks Easy -1.8
Three of the following points lie on the same straight line. Which point does not lie on this line? Tick one box. [1 mark] \((-2, 14)\) \((-1, 8)\) \((1, -1)\) \((2, -6)\)
AQA AS Paper 1 Specimen Q7
4 marks Moderate -0.8
Determine whether the line with equation \(2x + 3y + 4 = 0\) is parallel to the line through the points with coordinates \((9, 4)\) and \((3, 8)\). [4 marks]
Edexcel AS Paper 1 Q4
Moderate -0.8
The line \(l_1\) has equation \(2x - 3y = 9\) The line \(l_2\) passes through the points \((3, -1)\) and \((-1, 5)\) Determine, giving full reasons for your answer, whether lines \(l_1\) and \(l_2\) are parallel, perpendicular or neither.
WJEC Unit 1 Specimen Q5
12 marks Moderate -0.8
The points \(A(0, 2)\), \(B(-2, 8)\), \(C(20, 12)\) are the vertices of the triangle \(ABC\). The point \(D\) is the mid-point of \(AB\).
  1. Show that \(CD\) is perpendicular to \(AB\). [6]
  2. Find the exact value of \(\tan CAB\). [5]
  3. Write down the geometrical name for the triangle \(ABC\). [1]