Rectangle or parallelogram vertices - Straight Lines & Coordinate Geometry

CAIE P1 2005 June Q5

6 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-2_591_1061_1098_541} The diagram shows a rhombus $A B C D$. The points $B$ and $D$ have coordinates $( 2,10 )$ and $( 6,2 )$ respectively, and $A$ lies on the $x$-axis. The mid-point of $B D$ is $M$. Find, by calculation, the coordinates of each of $M , A$ and $C$.

CAIE P1 2007 June Q6

7 marks Moderate -0.5

6 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-3_593_878_269_635} The diagram shows a rectangle $A B C D$. The point $A$ is $( 2,14 ) , B$ is $( - 2,8 )$ and $C$ lies on the $x$-axis. Find

the equation of $B C$,
the coordinates of $C$ and $D$.

CAIE P1 2010 June Q8

9 marks Standard +0.3

8 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726} The diagram shows a rhombus $A B C D$ in which the point $A$ is ( $- 1,2$ ), the point $C$ is ( 5,4 ) and the point $B$ lies on the $y$-axis. Find

the equation of the perpendicular bisector of $A C$,
the coordinates of $B$ and $D$,
the area of the rhombus.

CAIE P1 2014 June Q11

9 marks Standard +0.8

11 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639} The diagram shows a parallelogram $A B C D$, in which the equation of $A B$ is $y = 3 x$ and the equation of $A D$ is $4 y = x + 11$. The diagonals $A C$ and $B D$ meet at the point $E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)$. Find, by calculation, the coordinates of $A , B , C$ and $D$.

CAIE P1 2018 June Q5

7 marks Moderate -0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-08_558_785_258_680} The diagram shows a kite $O A B C$ in which $A C$ is the line of symmetry. The coordinates of $A$ and $C$ are $( 0,4 )$ and $( 8,0 )$ respectively and $O$ is the origin.

Find the equations of $A C$ and $O B$.
Find, by calculation, the coordinates of $B$.

CAIE P1 2019 June Q4

6 marks Standard +0.3

4 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-06_625_750_260_699} The diagram shows a trapezium $A B C D$ in which the coordinates of $A , B$ and $C$ are (4, 0), (0, 2) and $( h , 3 h )$ respectively. The lines $B C$ and $A D$ are parallel, angle $A B C = 90 ^ { \circ }$ and $C D$ is parallel to the $x$-axis.

Find, by calculation, the value of $h$.
Hence find the coordinates of $D$.

CAIE P1 2002 November Q9

9 marks Moderate -0.3

9 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_719_958_264_589} The diagram shows a rectangle $A B C D$, where $A$ is $( 3,2 )$ and $B$ is $( 1,6 )$.

Find the equation of $B C$. Given that the equation of $A C$ is $y = x - 1$, find
the coordinates of $C$,
the perimeter of the rectangle $A B C D$.

CAIE P1 2003 November Q5

7 marks Moderate -0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-2_594_778_1360_682} The diagram shows a trapezium $A B C D$ in which $B C$ is parallel to $A D$ and angle $B C D = 90 ^ { \circ }$. The coordinates of $A , B$ and $D$ are $( 2,0 ) , ( 4,6 )$ and $( 12,5 )$ respectively.

Find the equations of $B C$ and $C D$.
Calculate the coordinates of $C$.

CAIE P1 2009 November Q9

11 marks Standard +0.8

9 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_554_723_1557_712} The diagram shows a rectangle $A B C D$. The point $A$ is $( 0 , - 2 )$ and $C$ is $( 12,14 )$. The diagonal $B D$ is parallel to the $x$-axis.

Explain why the $y$-coordinate of $D$ is 6 . The $x$-coordinate of $D$ is $h$.
Express the gradients of $A D$ and $C D$ in terms of $h$.
Calculate the $x$-coordinates of $D$ and $B$.
Calculate the area of the rectangle $A B C D$.

CAIE P1 2012 November Q5

6 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-3_602_751_255_696} The diagram shows a triangle $A B C$ in which $A$ has coordinates ( 1,3 ), $B$ has coordinates ( 5,11 ) and angle $A B C$ is $90 ^ { \circ }$. The point $X ( 4,4 )$ lies on $A C$. Find

the equation of $B C$,
the coordinates of $C$.

CAIE P1 2013 November Q5

7 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667} The diagram shows a rectangle $A B C D$ in which point $A$ is ( 0,8 ) and point $B$ is ( 4,0 ). The diagonal $A C$ has equation $8 y + x = 64$. Find, by calculation, the coordinates of $C$ and $D$.

CAIE P3 2018 June Q5

7 marks Moderate -0.5

5 \includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-08_558_785_258_680} The diagram shows a kite $O A B C$ in which $A C$ is the line of symmetry. The coordinates of $A$ and $C$ are $( 0,4 )$ and $( 8,0 )$ respectively and $O$ is the origin.

Find the equations of $A C$ and $O B$.
Find, by calculation, the coordinates of $B$.

Edexcel P1 2020 January Q6

8 marks Standard +0.3

6. The line $l _ { 1 }$ has equation $3 x - 4 y + 20 = 0$ The line $l _ { 2 }$ cuts the $x$-axis at $R ( 8,0 )$ and is parallel to $l _ { 1 }$

Find the equation of $l _ { 2 }$, writing your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found. The line $l _ { 1 }$ cuts the $x$-axis at $P$ and the $y$-axis at $Q$.
Given that $P Q R S$ is a parallelogram, find
the area of $P Q R S$,
the coordinates of $S$.

Edexcel P1 2022 January Q8

9 marks Standard +0.3

8. The line $l _ { 1 }$ has equation $$2 x - 5 y + 7 = 0$$

Find the gradient of $l _ { 1 }$ Given that
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $M$.
Using algebra and showing all your working, find the coordinates of $M$.
(Solutions relying on calculator technology are not acceptable.) Given that the diagonals of a square $A B C D$ meet at $M$,
find the coordinates of the point $C$.

Edexcel C1 2014 January Q6

11 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6081d81b-51d2-4140-9834-71ef7fd700b0-12_650_885_255_603} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The straight line $l _ { 1 }$ has equation $2 y = 3 x + 7$ The line $l _ { 1 }$ crosses the $y$-axis at the point $A$ as shown in Figure 2.

1. State the gradient of $l _ { 1 }$
2. Write down the coordinates of the point $A$. Another straight line $l _ { 2 }$ intersects $l _ { 1 }$ at the point $B ( 1,5 )$ and crosses the $x$-axis at the point $C$, as shown in Figure 2. Given that $\angle A B C = 90 ^ { \circ }$,
find an equation of $l _ { 2 }$ in the form $a x + b y + c = 0$, where $a$, b and $c$ are integers. The rectangle $A B C D$, shown shaded in Figure 2, has vertices at the points $A , B , C$ and $D$.
Find the exact area of rectangle $A B C D$.

Edexcel C1 2018 June Q8

8 marks Moderate -0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-20_1063_1319_251_365} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the straight line $l _ { 1 }$ with equation $4 y = 5 x + 12$

State the gradient of $l _ { 1 }$ The line $l _ { 2 }$ is parallel to $l _ { 1 }$ and passes through the point $E ( 12,5 )$, as shown in Figure 2.
Find the equation of $l _ { 2 }$. Write your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be determined. The line $l _ { 2 }$ cuts the $x$-axis at the point $C$ and the $y$-axis at the point $B$.
Find the coordinates of
1. the point $B$,
2. the point $C$. The line $l _ { 1 }$ cuts the $y$-axis at the point $A$.
  The point $D$ lies on $l _ { 1 }$ such that $A B C D$ is a parallelogram, as shown in Figure 2.
Find the area of $A B C D$.

Edexcel M2 Q3

11 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The points $A ( 3,0 )$ and $B ( 0,4 )$ are two vertices of the rectangle $A B C D$, as shown in Fig. 2.

Write down the gradient of $A B$ and hence the gradient of $B C$. The point $C$ has coordinates $( 8 , k )$, where $k$ is a positive constant.
Find the length of $B C$ in terms of $k$. Given that the length of $B C$ is 10 and using your answer to part (b),
find the value of $k$,
find the coordinates of $D$.

CAIE P1 2014 November Q9

8 marks Moderate -0.3

\includegraphics{figure_9} The diagram shows a trapezium $ABCD$ in which $AB$ is parallel to $DC$ and angle $BAD$ is $90°$. The coordinates of $A$, $B$ and $C$ are $(2, 6)$, $(5, -3)$ and $(8, 3)$ respectively.

Find the equation of $AD$. [3]
Find, by calculation, the coordinates of $D$. [3]

The point $E$ is such that $ABCE$ is a parallelogram.

Find the length of $BE$. [2]

Edexcel C1 Q3

11 marks Moderate -0.8

\includegraphics{figure_2} The points $A(3, 0)$ and $B(0, 4)$ are two vertices of the rectangle $ABCD$, as shown in Fig. 2.

Write down the gradient of $AB$ and hence the gradient of $BC$. [3]

The point $C$ has coordinates $(8, k)$, where $k$ is a positive constant.

Find the length of $BC$ in terms of $k$. [2]

Given that the length of $BC$ is 10 and using your answer to part (b),

find the value of $k$, [4]
find the coordinates of $D$. [2]

Edexcel C1 Q6

11 marks Moderate -0.8

\includegraphics{figure_2} The points $A (3, 0)$ and $B (0, 4)$ are two vertices of the rectangle $ABCD$, as shown in Fig. 2.

Write down the gradient of $AB$ and hence the gradient of $BC$. [3]

The point $C$ has coordinates $(8, k)$, where $k$ is a positive constant.

Find the length of $BC$ in terms of $k$. [2]

Given that the length of $BC$ is 10 and using your answer to part (b),

find the value of $k$, [4]
find the coordinates of $D$. [2]

Edexcel C1 Q10

11 marks Moderate -0.5

\includegraphics{figure_1} The points $A(3, 0)$ and $B(0, 4)$ are two vertices of the rectangle $ABCD$, as shown in Fig. 1.

Write down the gradient of $AB$ and hence the gradient of $BC$. [3]

The point $C$ has coordinates $(8, k)$, where $k$ is a positive constant.

Find the length of $BC$ in terms of $k$. [2]

Given that the length of $BC$ is 10 and using your answer to part (b),

find the value of $k$, [4]
find the coordinates of $D$. [2]

AQA AS Paper 1 2021 June Q4

9 marks Moderate -0.3

$ABCD$ is a trapezium where $A$ is the point $(1, -2)$, $B$ is the point $(7, 1)$ and $C$ is the point $(3, 4)$ $DC$ is parallel to $AB$. $AD$ is perpendicular to $AB$.

1. Find the equation of the line $CD$. [2 marks]
2. Show that point $D$ has coordinates $(-1, 2)$ [3 marks]
1. Find the sum of the length of $AB$ and the length of $CD$ in simplified surd form. [2 marks]
2. Hence, find the area of the trapezium $ABCD$. [2 marks]