Triangle and parallelogram problems

OCR C4 2013 January Q8

7 marks Standard +0.3

8 The points $A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )$ and $D ( 1,6,3 )$ form a quadrilateral $A B C D$.

Show that $A B = A D$.
Find a vector equation of the line through $A$ and the mid-point of $B D$.
Show that $C$ lies on the line found in part (ii).
What type of quadrilateral is $A B C D$ ?

Edexcel Paper 1 Specimen Q4

5 marks Moderate -0.8

Relative to a fixed origin $O$,
the point $A$ has position vector $\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$,
the point $B$ has position vector $4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$,
and the point $C$ has position vector $2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }$.
Given that $A B C D$ is a parallelogram,
1. find the position vector of point $D$.
The vector $\overrightarrow { A X }$ has the same direction as $\overrightarrow { A B }$.
Given that $| \overrightarrow { A X } | = 10 \sqrt { 2 }$,
find the position vector of $X$.

AQA C4 2011 January Q8

14 marks Standard +0.3

8 The coordinates of the points $A$ and $B$ are $( 3 , - 2,4 )$ and $( 6,0,3 )$ respectively.
The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Calculate the acute angle between $\overrightarrow { A B }$ and the line $l _ { 1 }$, giving your answer to the nearest $0.1 ^ { \circ }$.
The point $D$ lies on $l _ { 1 }$ where $\lambda = 2$. The line $l _ { 2 }$ passes through $D$ and is parallel to $A B$.
1. Find a vector equation of line $l _ { 2 }$ with parameter $\mu$.
2. The diagram shows a symmetrical trapezium $A B C D$, with angle $D A B$ equal to angle $A B C$. \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point $C$ lies on line $l _ { 2 }$. The length of $A D$ is equal to the length of $B C$. Find the coordinates of $C$.

AQA C4 2012 January Q8

12 marks Standard +0.3

8 The points $A$ and $B$ have coordinates $( 4 , - 2,3 )$ and $( 2,0 , - 1 )$ respectively. The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 2 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 5 \\ - 2 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Find the acute angle between $A B$ and the line $l$, giving your answer to the nearest degree.
The point $C$ lies on the line $l$ such that the angle $A B C$ is a right angle. Given that $A B C D$ is a rectangle, find the coordinates of the point $D$.

AQA C4 2013 January Q6

15 marks Standard +0.3

6

The points $A , B$ and $C$ have coordinates $( 3,1 , - 6 ) , ( 5 , - 2,0 )$ and $( 8 , - 4 , - 6 )$ respectively.
1. Show that the vector $\overrightarrow { A C }$ is given by $\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]$, where $n$ is an integer.
2. Show that the acute angle $A C B$ is given by $\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)$.
Find a vector equation of the line $A C$.
The point $D$ has coordinates $( 6 , - 1 , p )$. It is given that the lines $A C$ and $B D$ intersect.
1. Find the value of $p$.
2. Show that $A B C D$ is a rhombus, and state the length of each of its sides.

AQA C4 2010 June Q7

12 marks Standard +0.3

7 The point $A$ has coordinates $( 4 , - 3,2 )$.
The line $l _ { 1 }$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 3 \\ 2 \end{array} \right] + \lambda \left[ \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right]$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ - 2 \\ - 1 \end{array} \right]$.
The point $B$ lies on $l _ { 2 }$ where $\mu = 2$.

Find the vector $\overrightarrow { A B }$.
1. Show that the lines $l _ { 1 }$ and $l _ { 2 }$ intersect.
2. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$. Find the coordinates of $P$.
The point $C$ lies on a line which is parallel to $l _ { 1 }$ and which passes through the point $B$. The points $A , B , C$ and $P$ are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point $C$.

AQA C4 2016 June Q6

15 marks Challenging +1.2

6 The line $l _ { 1 }$ passes through the point $A ( 0,6,9 )$ and the point $B ( 4 , - 6 , - 11 )$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } - 1 \\ 5 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 5 \\ 1 \end{array} \right]$.

The acute angle between the lines $l _ { 1 }$ and $l _ { 2 }$ is $\theta$. Find the value of $\cos \theta$ as a fraction in its lowest terms.
Show that the lines $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of the point of intersection.
The points $C$ and $D$ lie on line $l _ { 2 }$ such that $A C B D$ is a parallelogram. \includegraphics[max width=\textwidth, alt={}, center]{c42685e9-bfa4-48d4-8abb-13e88a4b765e-12_392_949_1018_548} The length of $A B$ is three times the length of $C D$.
Find the coordinates of the points $C$ and $D$.
[0pt] [5 marks] $7 \quad$ A curve $C$ is defined by the parametric equations $$x = \frac { 4 - \mathrm { e } ^ { 2 - 6 t } } { 4 } , \quad y = \frac { \mathrm { e } ^ { 3 t } } { 3 t } , \quad t \neq 0$$

Edexcel C4 Q7

15 marks Standard +0.3

7. The line $l _ { 1 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 4 \end{array} \right)$$ and the line $l _ { 2 }$ has vector equation $$\mathbf { r } = \left( \begin{array} { r } 0 \\ 4 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) ,$$ where $\lambda$ and $\mu$ are parameters.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $B$ and the acute angle between $l _ { 1 }$ and $l _ { 2 }$ is $\theta$.

Find the coordinates of $B$.
Find the value of $\cos \theta$, giving your answer as a simplified fraction. The point $A$, which lies on $l _ { 1 }$, has position vector $\mathbf { a } = 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$.
The point $C$, which lies on $l _ { 2 }$, has position vector $\mathbf { c } = 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k }$.
The point $D$ is such that $A B C D$ is a parallelogram.
Show that $| \overrightarrow { A B } | = | \overrightarrow { B C } |$.
Find the position vector of the point $D$.

AQA C4 2007 January Q6

13 marks Moderate -0.3

6 The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 5,4,0 )$ and $( 11,6 , - 4 )$ respectively.

1. Find the vector $\overrightarrow { B A }$.
2. Show that the size of angle $A B C$ is $\cos ^ { - 1 } \left( - \frac { 5 } { 7 } \right)$.
The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } 8 \\ - 3 \\ 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right]$.
1. Verify that $C$ lies on $l$.
2. Show that $A B$ is parallel to $l$.
The quadrilateral $A B C D$ is a parallelogram. Find the coordinates of $D$.

AQA C4 2009 June Q7

10 marks Moderate -0.3

7 The points $A$ and $B$ have coordinates ( $3 , - 2,5$ ) and ( $4,0,1$ ) respectively. The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 6 \\ - 1 \\ 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 4 \end{array} \right]$.

Find the distance between the points $A$ and $B$.
Verify that $B$ lies on $l _ { 1 }$.
(2 marks)
The line $l _ { 2 }$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1 \\ 3 \\ - 8 \end{array} \right]$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $C$. Show that the points $A , B$ and $C$ form an isosceles triangle.
(6 marks)

AQA Paper 2 2018 June Q14

6 marks Moderate -0.3

14 A quadrilateral has vertices $A , B , C$ and $D$ with position vectors given by $$\overrightarrow { O A } = \left[ \begin{array} { l } 3 \\ 5 \\ 1 \end{array} \right] , \overrightarrow { O B } = \left[ \begin{array} { r } - 1 \\ 2 \\ 7 \end{array} \right] , \overrightarrow { O C } = \left[ \begin{array} { l } 0 \\ 7 \\ 6 \end{array} \right] \text { and } \overrightarrow { O D } = \left[ \begin{array} { r } 4 \\ 10 \\ 0 \end{array} \right]$$ 14

Write down the vector $\overrightarrow { A B }$ 14
Show that $A B C D$ is a parallelogram, but not a rhombus.