AQA Level 2 Further Maths · Level 2 Certificate · Chapter 5: Coordinate geometry · Lesson 3

Dividing a line in a given ratio

About 80 minutesNo calculator71 marks

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Date

Do Now

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

A1
Needed today
$M$ is the midpoint of $A B$ . $A$ is $(- 2, 7)$ and $M$ is $(3, 1)$ . Work out the coordinates of $B$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A2
Needed today
Work out the number that is $\frac{3}{8}$ of the way from $- 6$ to $14$ on a number line.
[2 marks]
Answer
A3
Last lesson
Work out the coordinates of the point where the lines $2 x + y = 7$ and $x - 3 y = 14$ meet.
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A4
From chapter 4
Solve $2 x^{2} + x - 15 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Example, then your turn

Level 2

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

Example 1

$A$ is $(- 3, 5)$ and $B$ is $(9, - 11)$ . $C$ is the point on $A B$ with $A C : C B = 1 : 3$ . Work out the coordinates of $C$ .

Not drawn accurately

Hint The ratio has

1 + 3 = 4

parts, so

C

\frac{1}{4}

of the way from

A

B

B1
$A$ is $(- 5, - 2)$ and $B$ is $(10, 8)$ . $C$ is the point on $A B$ with $A C : C B = 2 : 3$ . Work out the coordinates of $C$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Example 2

$A B C$ is a straight line. $A$ is $(1, 4)$ and $B$ is $(4, 2)$ . $A B : B C = 2 : 3$ . Work out the coordinates of $C$ .

Not drawn accurately

Hint

C

is beyond

B

B C

\frac{3}{2}

A B

, and goes the same way.

B2
$A B C$ is a straight line. $A$ is $(- 2, - 5)$ and $B$ is $(1, 1)$ . $A B : B C = 3 : 2$ . Work out the coordinates of $C$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Example 3

$A B C$ is a straight line. $A$ is $(- 1, 2)$ and $C$ is $(13, 9)$ . $A C : B C = 7 : 3$ . Work out the coordinates of $B$ .

Not drawn accurately

Hint

A C

is the whole line,

7

parts;

B C

3

of them. So

A B

4

parts and

B

\frac{4}{7}

of the way from

A

B3
$A B C$ is a straight line. $A$ is $(- 4, - 3)$ and $C$ is $(14, 15)$ . $A C : B C = 9 : 4$ . Work out the coordinates of $B$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Example 4

$C$ is the point on $A B$ with $A C : C B = 2 : 3$ . $A$ is $(- 1, 6)$ and $C$ is $(3, 2)$ . Work out the coordinates of $B$ .

Not drawn accurately

Hint

A

C

2

parts. One part is half of it.

C

B

3

parts.

B4
$C$ is the point on $A B$ with $A C : C B = 3 : 1$ . $A$ is $(5, - 7)$ and $C$ is $(- 4, 5)$ . Work out the coordinates of $B$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Example 5

$A$ is $(- 2, - 1)$ and $B$ is $(10, 5)$ . The point $C (k, 3)$ lies on $A B$ . Work out $k$ and the ratio $A C : C B$ .

Not drawn accurately

Hint Use the

y

-coordinates: from

- 1

5

6

; to reach

3

4

of those

6

B5
$A$ is $(- 3, 7)$ and $B$ is $(9, - 2)$ . The point $C (k, 4)$ lies on $A B$ . Work out the value of $k$ and the ratio $A C : C B$ .
[4 marks]
Answer $k = \dots\dots A C : C B = \dots\dots$

Practice

Level 2

C1
$D E F$ is a straight line. $E$ is $(5, - 1)$ and $F$ is $(13, 11)$ . $D E : E F = 3 : 4$ . Work out the coordinates of $D$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
C2
$A B C$ is a straight line. $A B$ is $50%$ longer than $B C$ . $A$ is $(- 6, 9)$ and $B$ is $(3, - 3)$ . (a) Work out the ratio $A B : B C$ in its simplest form. (b) Work out the coordinates of $C$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $(\dots\dots, \dots\dots)$

C3
The line $5 x + 2 y = 20$ meets the $x$ -axis at $A$ and the $y$ -axis at $B$ . $P$ is the point on $A B$ with $A P : P B = 3 : 1$ . Work out the coordinates of $P$ .
[3 marks]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$
C4
$P QR$ is a straight line. $P$ is $(- 2, a)$ , $Q$ is $(b, 4)$ and $R$ is $(10, 13)$ . $P Q : QR = 1 : 3$ . Work out the values of $a$ and $b$ .
[4 marks]
Answer $a = \dots\dots b = \dots\dots$
C5
$A$ is $(- 4, 1)$ , $B$ is $(2, 4)$ and $C$ is $(10, 8)$ . (a) Show that $A$ , $B$ and $C$ lie on a straight line. (b) Work out the ratio $A B : B C$ in its simplest form.
[3 marks]
Answer(b) $\dots\dots$

Exam-style questions

AQA exam style

D1
$A$ is $(0, 0)$ and $B$ is $(20, 15)$ . $C$ is the point on $A B$ with $A C : C B = 1 : 4$ . Circle the coordinates of $C$ . $(4, 3) (5, 3.75) (10, 7.5) (16, 12)$
[1 mark]
Answer

D2
$A$ is $(2, - 3)$ and $B$ is $(14, 6)$ . $C$ is on $A B$ with $A C : C B = 2 : 1$ . Jamal says $C$ is $(6, 0)$ , because $\frac{1}{3}$ of $12$ is $4$ and $\frac{1}{3}$ of $9$ is $3$ .
Explain Jamal's mistake. Work out the correct coordinates of $C$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
D3
$A B C$ is a straight line with $A B : B C = 3 : 5$ . $B$ is $(4, 6)$ and $C$ is $(14, 1)$ . Work out the coordinates of $A$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$
D4
The line $L$ has equation $x + 2 y = 16$ . $L$ meets the $y$ -axis at $P$ and the $x$ -axis at $Q$ . $R$ is the point on $P Q$ with $P R : R Q = 1 : 3$ . The line through $R$ perpendicular to $L$ meets the $x$ -axis at $S$ . Work out the coordinates of $S$ . You must show your working.
[5 marks]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$
D5
Triangle $A B C$ has vertices $A (- 1, 3)$ , $B (9, 8)$ and $C (4, 13)$ . $P$ is on $A B$ with $A P : P B = 2 : 3$ . $Q$ is on $A C$ with $A Q : QC = 2 : 3$ .
Not drawn accurately
(a)
Work out the coordinates of $P$ and of $Q$ .
[2 marks]
Answer $P (\dots, \dots) Q (\dots, \dots)$
(b)
Show that $P Q$ is parallel to $B C$ .
[2 marks]
(c)
The area of triangle $A B C$ is $37.5$ . Work out the area of the quadrilateral $P B C Q$ .
[3 marks]
Answer
Total for D5: 7 marks
D6
$P$ is $(1, 2)$ and $Q$ is $(9, 14)$ . $R$ lies on $P Q$ and on the line $y = 11$ . $P R : R Q = k : 1$ . Work out the value of $k$ and the coordinates of $R$ .
[4 marks]
Answer $k = \dots\dots R (\dots\dots, \dots\dots)$

Extension

Stretch

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

E1
Triangle $A B C$ has vertices $A (- 3, - 3)$ , $B (9, 0)$ and $C (0, 9)$ . $P$ is on $A B$ with $A P : P B = 1 : 2$ , $Q$ is on $B C$ with $B Q : QC = 1 : 2$ and $R$ is on $C A$ with $C R : R A = 1 : 2$ . Show that the area of triangle $P QR$ is one third of the area of triangle $A B C$ .
[6 marks]
Not drawn accurately
Hint 1 · What to try
Find $P$ , $Q$ and $R$ first. Then find both areas: put each triangle in a rectangle and take off the right-angled triangles round it.
Hint 2 · The first line
$P$ is $\frac{1}{3}$ of the way from $A$ to $B$ : $(1, - 2)$ . In the same way $Q$ is $(6, 3)$ and $R$ is $(- 1, 5)$ .
Hint 3 · The full method
$A B C$ sits in the $12$ by $12$ square from $(- 3, - 3)$ : $144 - 18 - 40.5 - 18 = 67.5$ . $P QR$ sits in the $7$ by $7$ square from $(- 1, - 2)$ : $49 - 12.5 - 7 - 7 = 22.5$ . And $22.5 = \frac{1}{3} \times 67.5$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the point that divides a line segment in a given ratio.
I can find a point beyond the end of a line, or the far end, from a ratio.
I can work out the ratio in which a point divides a line from its coordinates.
I can use a dividing point in a longer problem, with lines and areas.

Answers and mark scheme

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