AQA Level 2 Further Maths · Level 2 Certificate · Chapter 5: Coordinate geometry · Lesson 1

Parallel and perpendicular lines, distance and midpoint

About 90 minutesNo calculator70 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-parallel-and-perpendicular-lines-distance-and-midpoint/

Date

Do Now

The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out the gradient of the line through the points $(- 2, 5)$ and $(4, - 7)$ .
[2 marks]
Answer
A2
Needed today
Write $80$ in the form $a 5$ , where $a$ is an integer.
[2 marks]
Answer $a = \dots\dots$
A3
Last lesson
$x + y + z = 9$ , $y = 2 x$ and $z = x + 1$ . Work out the values of $x$ , $y$ and $z$ .
[2 marks]
Answer $x = \dots\dots y = \dots\dots z = \dots\dots$
A4
From chapter 4
Work out the value of $2 7^{- \frac{2}{3}}$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

$A$ is the point $(- 2, - 3)$ and $B$ is the point $(10, 2)$ . Work out (a) the gradient of a line perpendicular to $A B$ , (b) the length of $A B$ , (c) the coordinates of the midpoint of $A B$ .

Not drawn accurately

Hint Gradient of

A B

is the change in

y

over the change in

x

. A perpendicular gradient is

- 1

over it. For the length, use Pythagoras on the two changes.

B1
$P$ is $(1, 7)$ and $Q$ is $(- 5, 11)$ . Work out the gradient of a line perpendicular to $P Q$ , the exact length of $P Q$ in its simplest form, and the midpoint of $P Q$ .
[5 marks]
Not drawn accurately

Example 2

$M (3, - 1)$ is the midpoint of the line $A B$ . $A$ is the point $(- 4, 5)$ . Work out the coordinates of $B$ .

Hint To get from

A

M

you go some way in

x

and in

y

. Go the same again from

M

to reach

B

B2
$M (2, - 3)$ is the midpoint of the line $A B$ , where $A$ is $(p, 4)$ and $B$ is $(7, r)$ . Work out the values of $p$ and $r$ .
[2 marks]
Answer $p = \dots\dots r = \dots\dots$

Example 3

$A (1, 1)$ , $B (7, 3)$ and $C (5, 9)$ are the vertices of a triangle. Show that the triangle is right-angled (a) by using gradients, (b) by using lengths.

Not drawn accurately

Hint Find the gradients of the three sides: two that multiply to

- 1

meet at a right angle. For lengths, test

a^{2} + b^{2} = c^{2}

with the longest side as

c

B3
$D (- 2, 1)$ , $E (2, 3)$ and $F (- 1, 9)$ are the vertices of a triangle. Show that the triangle is right-angled.
[3 marks]
Not drawn accurately

Example 4

$A$ is $(1, 1)$ , $B$ is $(3, b)$ and $C$ is $(7, 3)$ . Angle $A B C = 9 0^{\circ}$ . Work out the two possible values of $b$ .

Hint Write the gradients of

A B

and

B C

in terms of

b

. A right angle at

B

means they multiply to

- 1

B4
$X$ is $(- 2, 3)$ , $Y$ is $(1, a)$ and $Z$ is $(5, 2)$ . Angle $X Y Z = 9 0^{\circ}$ . Work out the two possible values of $a$ .
[4 marks]
Answer $a = \dots\dots$ or $a = \dots\dots$

Example 5

$W X Y Z$ is a parallelogram. $W$ is $(1, 2)$ , $X$ is $(6, 3)$ and $Y$ is $(8, 7)$ . Work out the coordinates of $Z$ .

Not drawn accurately

Hint The diagonals of a parallelogram cut each other in half, so

W Y

and

X Z

have the same midpoint.

B5
$A$ is $(- 1, 2)$ , $B$ is $(3, 0)$ and $C$ is $(5, 4)$ . Show that angle $A B C = 9 0^{\circ}$ . Hence work out the coordinates of $D$ such that $A B C D$ is a rectangle.
[4 marks]
Not drawn accurately
Answer $D = (\dots\dots, \dots\dots)$

Practice

Level 2

C1
$P$ is the point $(2 a, b)$ and $Q$ is the point $(6 a, 7 b)$ , where $a$ and $b$ are positive. Write, in terms of $a$ and $b$ , (a) the gradient of $P Q$ , (b) the length of $P Q$ , (c) the midpoint of $P Q$ .
[4 marks]
C2
The distance between the points $A (3, k)$ and $B (- 5, 1)$ is $17$ . Work out the two possible values of $k$ .
[3 marks]
Answer $k = \dots\dots$ or $k = \dots\dots$

C3
$P (- 1, 2)$ , $Q (- 1, 10)$ and $R (5, 4)$ are the vertices of a triangle. Work out the area of the triangle and its exact perimeter, giving the perimeter with each surd in its simplest form.
[4 marks]
Not drawn accurately
Answerarea $\dots\dots$ perimeter $\dots\dots\dots\dots$
C4
The points $P (- 3, 1)$ , $Q (1, q)$ and $R (9, 7)$ lie on a straight line. Work out the value of $q$ .
[2 marks]
Answer $q = \dots\dots$
C5
$A$ is $(0, 3)$ , $B$ is $(4, 5)$ , $C$ is $(8, 1)$ and $D$ is $(0, - 3)$ . Show that $A B C D$ is a trapezium.
[3 marks]

Exam-style questions

AQA exam style

D1
Work out the distance between the points $P (- 4, 9)$ and $Q (3, - 15)$ .
[2 marks]
Answer $\dots\dots$ units

D2
Ali says, "The line $L$ has gradient $- 2$ , so a line perpendicular to $L$ has gradient $2$ ."
The line $L$ passes through the points $(- 1, 6)$ and $(3, - 2)$ .
Not drawn accurately
(a)
Work out the gradient of a line perpendicular to $L$ .
[2 marks]
Answer
(b)
Explain what Ali has done wrong.
[1 mark]
Total for D2: 3 marks
D3
$A$ is $(- 2, 1)$ , $B$ is $(2, 4)$ , $C$ is $(7, 4)$ and $D$ is $(3, 1)$ .
Not drawn accurately
(a)
Show that $A B C D$ is a rhombus.
[3 marks]
(b)
Show that the diagonals $A C$ and $B D$ are perpendicular.
[2 marks]
(c)
Work out the area of $A B C D$ .
[2 marks]
Answer
Total for D3: 7 marks
D4
$A$ is $(1, - 2)$ , $B$ is $(9, 2)$ and $C$ is $(2, 6)$ .
Not drawn accurately
(a)
Show that triangle $A B C$ is isosceles.
[3 marks]
(b)
$M$ is the midpoint of $A B$ . Work out the coordinates of $M$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(c)
Hence work out the area of triangle $A B C$ .
[3 marks]
Answer
Total for D4: 7 marks
D5
The point $P (k, 2 k)$ is $5$ units from the point $Q (1, - 3)$ . Work out the two possible coordinates of $P$ . You must show your working.
[4 marks]
Answer $(\dots\dots, \dots\dots)$ or $(\dots\dots, \dots\dots)$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$A (1, 1)$ and $B (5, 3)$ are two corners of a square. Work out every possible pair of positions for the other two corners.
[5 marks]
Hint 1 · What to try
$A B$ could be a side of the square (on either side of it) or a diagonal. Use the step from $A$ to $B$ .
Hint 2 · The first line
From $A$ to $B$ is $4$ across and $2$ up. A step at right angles is $2$ back and $4$ up, or $2$ across and $4$ down.
Hint 3 · The full method
As a side: add $(- 2, 4)$ to $A$ and $B$ , or add $(2, - 4)$ . As a diagonal: the centre is $(3, 2)$ and the half-diagonal $(2, 1)$ turned a right angle is $(- 1, 2)$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can tell if two lines are parallel or perpendicular from their gradients.
I can work out the exact length and the midpoint of a line between two points.
I can find an unknown coordinate from a right angle, a distance or a midpoint.
I can prove facts about triangles and quadrilaterals using coordinates.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers