AQA Level 2 Further Maths · Level 2 Certificate · Chapter 7: Geometry II

Area of a triangle, the sine and cosine rules, and problems in three dimensions

About 90 minutesCalculator allowed122 marks

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Date

Do Now

The first one is what today needs. The others bring back earlier work.

A1
Needed today
Write down the exact value of $sin 12 0^{\circ}$ and the exact value of $cos 15 0^{\circ}$ .
[2 marks]
Answer $sin 12 0^{\circ} = \dots\dots$ $cos 15 0^{\circ} = \dots\dots$

A2
From chapter 6
Solve $5 cos x + 2 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give your answers to 1 decimal place.
[2 marks]
Answer
A3
From chapter 4
Solve $x^{2} - 3 x - 54 = 0$ .
[2 marks]
Answer
A4
GCSE Higher
In triangle $P QR$ , angle $P QR = 9 0^{\circ}$ , $P Q = 6.4$ cm and $QR = 9.1$ cm. Work out the size of angle $QP R$ . Give your answer to 1 decimal place.
[2 marks]
Answer $\dots \dots^{\circ}$
A5
From chapter 5
Work out the distance between the points $(- 1, 4)$ and $(5, - 2)$ . Give your answer as a surd in its simplest form.
[2 marks]
Answer

Examples, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it on your own. Give lengths and angles to 3 significant figures.

Example 1

In triangle $A B C$ , $A B = 12$ cm, $A C = 9$ cm and angle $A B C = 3 4^{\circ}$ .

Work out the two possible lengths of $B C$ .

Hint The sine rule gives

sin C = \frac{12 sin 3 4 ^{\circ}}{9}

. The calculator gives the acute angle;

18 0^{\circ}

minus it also works, as it still leaves room in the triangle. Find angle

A

for each, then use the sine rule again for

B C

B1
In triangle $P QR$ , $P Q = 14$ cm, $P R = 10$ cm and angle $P QR = 4 1^{\circ}$ .
Work out the two possible lengths of $QR$ .
[4 marks]
Answer $QR = \dots\dots$ cm or $\dots\dots$ cm

Example 2

In triangle $A B C$ , angle $B A C = 12 0^{\circ}$ , $A C = 11$ cm and $B C = 31$ cm. $A B = x$ cm.

Work out the value of $x$ .

Not drawn accurately

Hint Use the cosine rule for

B C

, the side opposite

12 0^{\circ}

3 1^{2} = x^{2} + 1 1^{2} - 2 \times 11 x cos 12 0^{\circ}

. As

cos 12 0^{\circ} = - \frac{1}{2}

, this is

x^{2} + 11 x - 840 = 0

. Factorise and keep the positive root.

B2
In triangle $A B C$ , angle $B A C = 12 0^{\circ}$ , $A C = 13$ cm and $B C = 43$ cm. $A B = x$ cm.
Work out the value of $x$ .
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$

Example 3

In triangle $A B C$ , $A B = 7$ cm and $A C = 10$ cm. The area of the triangle is $28$ cm² and angle $B A C$ is obtuse.

Work out the length of $B C$ .

Not drawn accurately

Hint

\frac{1}{2} \times 7 \times 10 sin A = 28

, so

sin A = 0.8

. The obtuse angle is

18 0^{\circ} - 53. 1^{\circ} = 126. 9^{\circ}

. Then the cosine rule gives

B C

;

cos 126. 9^{\circ}

is negative, so

B C^{2}

is more than

7^{2} + 1 0^{2}

B3
In triangle $A B C$ , $A B = 9$ cm and $A C = 13$ cm. The area of the triangle is $35.1$ cm² and angle $B A C$ is obtuse.
Work out the length of $B C$ .
[4 marks]
Not drawn accurately
Answer $B C = \dots\dots$ cm

Example 4

$A B C D E F G H$ is a cuboid with $A B = 8$ cm, $B C = 6$ cm and $A E = 5$ cm. $N$ is the point on $A C$ such that $D N$ is perpendicular to $A C$ .

Work out the angle between the plane $A C H$ and the base $A B C D$ .

Not drawn accurately

Hint The angle is

H N D

, as

H N

and

D N

are both perpendicular to

A C

. In triangle

A D C

A C = 10

and the area is

\frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times 10 \times D N

, so

D N = 4.8

. Then

tan H N D = \frac{5}{4.8}

B4
$A B C D E F G H$ is a cuboid with $A B = 12$ cm, $B C = 5$ cm and $A E = 4$ cm. $N$ is the point on $A C$ such that $D N$ is perpendicular to $A C$ .
Work out the angle between the plane $A C H$ and the base $A B C D$ .
[4 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$

Practice

Level 2

Each question asks for something different. Show your working. Give lengths and angles to 3 significant figures unless the question says otherwise.

C1
A regular pentagon has sides of $8$ cm. Work out its area.
[4 marks]
Answer $\dots\dots$ cm²

C2
In triangle $A B C$ , $A B = 15$ cm, angle $C A B = 4 7^{\circ}$ and angle $A B C = 6 6^{\circ}$ .
Work out the perimeter of the triangle.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ cm
C3
In triangle $D E F$ , $D E = 12.5$ cm, $E F = 9.8$ cm and angle $D F E = 7 1^{\circ}$ .
Work out the size of angle $E D F$ . Explain why there is only one possible answer.
[3 marks]
Not drawn accurately
Answerangle $E D F = \dots \dots^{\circ}$
C4
In triangle $X Y Z$ , $X Y = 7.3$ cm, $X Z = 11.6$ cm and angle $Y X Z = 12 8^{\circ}$ . Work out the length of $Y Z$ .
[3 marks]
Not drawn accurately
Answer $Y Z = \dots\dots$ cm
C5
Triangle $A B C$ has $A B = 15$ cm, $A C = 8$ cm and $B C = 13$ cm.
(a)
Show that angle $B A C = 6 0^{\circ}$ .
[3 marks]
(b)
Work out the exact area of the triangle.
[2 marks]
Answer $\dots\dots$ cm²
Total for C5: 5 marks
C6
$P QR S$ is a quadrilateral. $P Q = 5.4$ cm, $QR = 8.2$ cm, angle $P QR = 9 6^{\circ}$ , angle $P R S = 3 8^{\circ}$ and angle $R P S = 5 7^{\circ}$ .
Not drawn accurately
(a)
Work out the length of $P R$ .
[2 marks]
Answer $P R = \dots\dots$ cm
(b)
Work out the length of $R S$ .
[2 marks]
Answer $R S = \dots\dots$ cm
Total for C6: 4 marks
C7
$A B C D E F G H$ is a cuboid with $A B = 10$ cm, $B C = 4$ cm and $A E = 6$ cm.
Not drawn accurately
(a)
Work out the length of the diagonal $B H$ .
[2 marks]
Answer $B H = \dots\dots$ cm
(b)
Work out the angle between $B H$ and the base $A B C D$ .
[2 marks]
Answer $\dots \dots^{\circ}$
Total for C7: 4 marks
C8
$P$ , $Q$ and $F$ are on level ground. $F T$ is a vertical mast. $P Q = 85$ m, $P F = 60$ m and angle $F P Q = 4 8^{\circ}$ . The angle of elevation of $T$ from $P$ is $2 6^{\circ}$ .
Work out the angle of elevation of $T$ from $Q$ .
[5 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$
C9
A cone has a slant height of $13$ cm. The slant edge makes an angle of $6 2^{\circ}$ with the base. Work out the volume of the cone.
(The volume of a cone is $\frac{1}{3} π r^{2} h$ .)
[3 marks]
Not drawn accurately
Answer $\dots\dots$ cm³

Exam-style questions

AQA exam style

Answer every question in the space given. Show your working: most of the marks are for method.

D1
A triangle has sides of $6$ cm and $9$ cm with an angle of $15 0^{\circ}$ between them. Which of these is its area, in cm²? Circle your answer.
$13.5 27 13.5 3273$
[1 mark]
Answer

D2
Leo says, "A triangle with sides $4$ cm, $6$ cm and $11$ cm has an obtuse angle, because $1 1^{2}$ is more than $4^{2} + 6^{2}$ ."
Use the cosine rule to show that Leo is wrong.
[3 marks]
D3
In triangle $A B C$ , $A B = 7$ cm, angle $C A B = 3 2^{\circ}$ and angle $A B C = 11 5^{\circ}$ .
Not drawn accurately
(a)
Work out the length of $B C$ .
[2 marks]
Answer $B C = \dots\dots$ cm
(b)
Work out the area of the triangle.
[2 marks]
Answer $\dots\dots$ cm²
Total for D3: 4 marks
D4
In triangle $A B C$ , $A C = 10$ cm and $B C = 14$ cm. The area of the triangle is $42$ cm².
(a)
Show that $sin A C B = 0.6$ .
[2 marks]
(b)
Hence work out the two possible lengths of $A B$ . Give your answers as surds in their simplest form.
[4 marks]
Answer $\dots\dots$ cm or $\dots\dots$ cm
Total for D4: 6 marks
D5
In triangle $A B C$ , $A C = (x + 1)$ cm, $A B = (2 x - 3)$ cm and angle $B A C = 15 0^{\circ}$ . The area of the triangle is $10.5$ cm².
Not drawn accurately
(a)
Show that $2 x^{2} - x - 45 = 0$ .
[3 marks]
(b)
Hence work out the length of $B C$ .
[3 marks]
Answer $B C = \dots\dots$ cm
Total for D5: 6 marks
D6
The diagram shows a ramp. The base $A B C D$ is a horizontal rectangle and the face $B C F E$ is vertical. $A B = 6.5$ m, $B C = 2.4$ m and $C F = 0.9$ m.
A ramp is allowed if its sloping face $A D F E$ makes an angle of less than $8^{\circ}$ with the horizontal.
Not drawn accurately
(a)
Is this ramp allowed? You must show your working.
[2 marks]
Answer $\dots \dots^{\circ}$ allowed? $\dots\dots$
(b)
Work out the length of $A F$ .
[2 marks]
Answer $A F = \dots\dots$ m
(c)
Work out the angle between $A F$ and the base $A B C D$ .
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D6: 6 marks
D7
$V A B C D$ is a pyramid. Its base $A B C D$ is a horizontal square of side $16$ cm. $V$ is directly above the centre of the base, and each sloping edge is $17$ cm long.
Not drawn accurately
(a)
Work out the size of angle $A V C$ .
[3 marks]
Answer $\dots \dots^{\circ}$
(b)
Hence work out the angle between $V A$ and the base.
[1 mark]
Answer $\dots \dots^{\circ}$
(c)
Work out the angle between the face $V B C$ and the base.
[3 marks]
Answer $\dots \dots^{\circ}$
Total for D7: 7 marks
D8
An orienteer runs $4.2$ km from $A$ to $B$ on a bearing of $06 5^{\circ}$ . She then runs $5.8$ km from $B$ to $C$ on a bearing of $17 0^{\circ}$ .
Not drawn accurately
(a)
Show that angle $A B C = 7 5^{\circ}$ .
[2 marks]
(b)
Work out the distance $A C$ .
[2 marks]
Answer $A C = \dots\dots$ km
(c)
Work out the bearing of $A$ from $C$ .
[3 marks]
Answer $\dots \dots^{\circ}$
(d)
Work out the area of the triangle $A B C$ .
[2 marks]
Answer $\dots\dots$ km²
Total for D8: 9 marks
D9
A lighthouse $L$ is $15$ km due north of a port $P$ . A ship sails from $P$ on a bearing of $03 0^{\circ}$ . It is exactly $9$ km from the lighthouse at two points, $S_{1}$ and $S_{2}$ , on its course.
Work out the distance between $S_{1}$ and $S_{2}$ .
[4 marks]
Not drawn accurately
Answer $\dots\dots$ km
D10
$A B$ is a chord of a circle, centre $O$ , radius $9$ cm. Angle $A O B = 10 0^{\circ}$ .
Work out the area of the shaded segment.
[3 marks]
Answer $\dots\dots$ cm²

Extension

Stretch

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A triangle has sides of $5$ cm, $6$ cm and $7$ cm. Work out its exact area.
[4 marks]
Hint 1 · What to try
Use the cosine rule for one angle, then $\frac{1}{2} ab sin C$ . Keep everything exact.
Hint 2 · The first line
The angle opposite the $7$ cm side: $cos C = \frac{25 + 36 - 49}{2 \times 5 \times 6} = \frac{1}{5}$ .
Hint 3 · The full method
$sin C = 1 - \frac{1}{25} = \frac{2 6}{5}$ , so the area is $\frac{1}{2} \times 5 \times 6 \times \frac{2 6}{5} = 66$ .
Answer $\dots\dots$ cm²

E2
$A B C D$ is a quadrilateral whose four vertices lie on a circle. $A B = 3$ cm, $B C = 5$ cm, $C D = 6$ cm and $D A = 8$ cm.
Work out the size of angle $A B C$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
Opposite angles of a cyclic quadrilateral add to $18 0^{\circ}$ , so $cos A D C = - cos A B C$ . The diagonal $A C$ is in two triangles.
Hint 2 · The first line
Write $A C^{2}$ twice: $3^{2} + 5^{2} - 30 cos B = 6^{2} + 8^{2} + 96 cos B$ .
Hint 3 · The full method
$34 - 30 cos B = 100 + 96 cos B$ , so $126 cos B = - 66$ , $cos B = - \frac{11}{21}$ and $B = 121. 6^{\circ}$ .
Answer $\dots \dots^{\circ}$
E3
$V A B C D$ is a pyramid on a horizontal square base, with $V$ directly above the centre $X$ of the base. $M$ is the midpoint of $B C$ . The edge $V B$ makes an angle $α$ with the base and the face $V B C$ makes an angle $β$ with the base.
Show that $tan β = 2 tan α$ .
[4 marks]
Hint 1 · What to try
Call the side of the base $2 a$ and the height $h$ . Both angles are in right-angled triangles with the vertical side $V X$ .
Hint 2 · The first line
$X B$ is half the diagonal, $a 2$ , so $tan α = \frac{h}{a 2}$ . $X M = a$ , so $tan β = \frac{h}{a}$ .
Hint 3 · The full method
$tan β = \frac{h}{a} = 2 \times \frac{h}{a 2} = 2 tan α$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the area of a triangle with $\frac{1}{2} ab sin C$ , and work back from an area to an angle or a side.
I can use the sine rule for a side or an angle, and I know when there are two possible triangles.
I can use the cosine rule for a side or an angle, including when it gives a quadratic, and use it with bearings.
I can find lengths and angles in three dimensions, including the angle between a line and a plane and between two planes.

Answers and mark scheme

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