Coordinate Geometry (Edexcel GCSE Maths: Higher): Exam Questions

$A$ is the point (—1, 2)
$B$ is the point (7, 5)

Find the coordinates of the midpoint of $AB$.

$P$ is the point (—4, 4)
$Q$ is the point (1, —5)

Find the gradient of $PQ$.

Point $P$ has coordinates (5, 7).
Point $M$ has coordinates (1, 2.5).

Point $M$ is the midpoint of the line $PQ$.

Find the coordinates of point $Q$.

Here are the equations of four straight lines.

Line A   $y=2x+4$
Line B  $2y=x+4$
Line C  $2x+2y=4$
Line D  $2x—y=4$

Two of these lines are parallel.

Write down the two parallel lines?

Point $A$ has coordinates $(–3,11)$
Point $B$ has coordinates $(47,b)$
The midpoint of $AB$ has coordinates $(a,–19)$

Find the value of $a$ and the value of $b$.

a = ................................................     
b = ................................................     

$P$ is the point (2, 14)
$Q$ is the point (6, 8)
$R$is the point (2, 5)

Use gradients to show that angle $PQR$ is not a right angle.

$A$ (0, 2) and $B$ (6, 5) are points on the straight line $ABCD$.

$AB=BC=CD$
Work out the coordinates of $D$.

$A$ is the point (2, –5)

$B$ is the point (4, –9)

Show that the gradient of the straight line passing through $A$ and $B$ is –2

$C$ is the point (–301, 601)

Does $C$lie on the straight line passing through $A$and $B$?

You must show your working.

$A,B$ and $C$ are points on the axes as shown.

The area of triangle $ABC$ is 28 square units.

Work out possible coordinates for $A,B$ and $C$.

$A$ ( ......... , .......... )

$B$ ( ......... , .......... )

$C$ ( ......... , .......... )

Here is a cuboid drawn on a 3-D grid.

$P$ is a vertex of the cuboid.

$T$ divides the line $OP$ in the ratio 1 : 2

Find the coordinates of $T$.

$A$ and $B$are two points.

Point $A$ has coordinates (—2, 4).
Point $B$ has coordinates (8, 9).

$C$ is the midpoint of the line segment $AB$.

Find the coordinates of $C$.

$D$ is the point with coordinates (100, 56).

Does point $D$ lie on the straight line that passes through $A$ and $B$?
You must show how you work out your answer.

The points $A,B$ and $C$ lie in order on a straight line.

The coordinates of $A$ are (2, 5)
The coordinates of $B$ are (4, $p$)
The coordinates of $C$ are ($q$, 17)

Given that $AC=4AB$, find the values of $p$ and $q$.

$Q,R$ and $S$ are points on a grid.

$Q$ is the point with coordinates (106, 103)
$R$ is the point with coordinates (106, 105)
$S$ is the point with coordinates (104, 105.5)

$P$ and $A$ are two other points on the grid such that

$R$ is the midpoint of $PQ$
$S$ is the midpoint of $PA$

Work out the coordinates of the point $A$.

A triangle has vertices $P,Q$ and $R$.

The coordinates of $P$ are (—3, —6)
The coordinates of $Q$ are (1, 4)
The coordinates of $R$ are (5, —2)

$M$ is the midpoint of $PQ$.
$N$ is the midpoint of $QR$.

Prove that $MN$ is parallel to $PR$.
You must show each stage of your working.

A pattern is made from four identical squares.

The sides of the squares are parallel to the axes.

Point $A$ has coordinates (6, 7)
Point $B$ has coordinates (38, 36)
Point $C$ is marked on the diagram.

Work out the coordinates of $C$.

$AB$ is a line segment.

$A$ is the point with coordinates (3, 6, 7).
The midpoint of $AB$ has coordinates (—2, 2, 5).

Find the coordinates of $B$.

$\mathbf{A}$ and $\mathbf{B}$ are straight lines.
Line $\mathbf{A}$ has equation $2y=3x+8$
Line $\mathbf{B}$ goes through the points $(—1,2)$ and $(2,8)$

Do lines $\mathbf{A}$ and $\mathbf{B}$ intersect?
You must show all your working.

A rectangle $ABCD$ is to be drawn on a centimetre grid such that

$A$ has coordinates (–4, –2)
$B$ has coordinates (1, 10)
$C$ has coordinates (19, $a$)
$D$ has coordinates $(b,c)$

Work out the value of $a$, the value of $b$ and the value of $c$.

$\begin{array}{l}a=.................................\\ b=.................................\\ c=.................................\end{array}$

Calculate the perimeter, in centimetres, of rectangle ABCD.

...................................................... cm

The diagram shows a line joining $O$ to $P$.

The gradient of the line is 2

The length of the line is $\sqrt{2645}$

Work out the coordinates of $P.$

A pattern is made from four congruent right-angled triangles.

The line AC is parallel to the x - axis.
The point A has coordinates (4, 9) and the point B has coordinates (17, 12).

Work out the coordinates of point C and point D.

$\begin{array}{l}C(...........................,..........................)\\ D(...........................,..........................)\end{array}$

A straight line goes through the points ($p,q$) and ($r,s$), where

• $p+2=r$

• $q+4=s.$

Find the gradient of the line.

$ABC$ is a triangle in which angle $ABC=90°$

$p$ and $q$ are integers such that

the coordinates of $A$ are $(p,10)$
the coordinates of $B$ are $(–1,–5)$
the coordinates of $C$are $(8,q)$

Given that the gradient of $AC$ is $-\frac{6}{7}$ work out the value of $p$ and the value of $q$

$p=$ .......................................................      
$q=$ .......................................................      

The straight line $\mathbf{L}$ has equation     $x–y=3$
The curve $\mathbf{C}$ has equation $3x^2–y^2+xy=9$

$\mathbf{L}$ and $\mathbf{C}$ intersect at the points $P$ and $Q$.

Find the coordinates of the midpoint of $PQ$.
Show clear algebraic working.

Triangle $HJK$ is isosceles with $HJ=HK$ and $JK=\sqrt{80}$

$H$ is the point with coordinates $(-4,1)$
$J$ is the point with coordinates $(j,15)$ where $j<0$
$K$ is the point with coordinates $(6,k)$

$M$is the midpoint of $JK$.
The gradient of $HM$ is 2

Find the value of $j$and the value of $k$.

The curve with equation $x^2–x+y^2=10$ and the straight line with equation $x–y=–4$ intersect at the points $A$and $B$.

Work out the exact length of $AB.$
Show your working clearly and give your answer in the form $\frac{\sqrt{a}}{2}$ where $a$ is an integer.

The curve with equation $y=(10x-3)(x+1)$ and the line with equation $y-6x=0$ intersect at the points $A$ and $B$.

Find the coordinates of the midpoint of $AB$. Show your working clearly.