A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 9: Trigonometric ratios · Lesson 2

Areas of triangles and solving triangle problems

72 marks

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Quick questions to start.

A1
Last lesson
A triangle has sides $5$ cm, $7$ cm and $8$ cm. Find the angle opposite the $7$ cm side.
[2 marks]
Answer
A2
Needed today
Solve $sin x = 0.35$ for $0^{\circ} < x < 18 0^{\circ}$ , to 1 d.p.
[2 marks]
Answer $x = \dots\dots$
A3
From chapter 8
Find the coefficient of $x^{3}$ in the expansion of $(2 + x)^{6}$ .
[2 marks]
Answer
A4
From chapter 2
Write $3 + 4 x - x^{2}$ in the form $p - (x - q)^{2}$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Find the area of triangle $A B C$ , to 3 s.f.

Hint Area

= \frac{1}{2} ab sin C

, where

C

is the angle between the two sides you know.

B1
Angle $P QR = 12 8^{\circ}$ . Find the area of triangle $P QR$ , to 3 s.f.
[2 marks]
Not drawn accurately
Answer

Example 2

Triangle $A B C$ has area $25$ cm² and angle $A B C$ is obtuse. Find angle $A B C$ and $A C$ .

Hint Put the area into

\frac{1}{2} a c sin B

and solve for

sin B

. Of the two angles, take the obtuse one. Then use the cosine rule.

B2
Triangle $X Y Z$ has area $54$ cm² and angle $Y X Z$ is acute. Find angle $Y X Z$ and $Y Z$ .
[4 marks]
Not drawn accurately
Answerangle $= \dots\dots$ $Y Z = \dots\dots$

Example 3

Find the area of this triangle, to 3 s.f.

Hint You need an angle first. Find any angle with the cosine rule, then use

\frac{1}{2} ab sin C

with the two sides beside it.

B3
Find the area of this triangle, to 3 s.f.
[4 marks]
Not drawn accurately
Answer

Example 4

$A$ , $B$ and the foot $F$ of a tower $F T$ are in a line on level ground. Find the height $F T$ .

Hint In triangle

A B T

the angle at

T

3 7^{\circ} - 1 8^{\circ}

. Use the sine rule for

B T

, then right-angled trigonometry in triangle

B F T

B4
$P$ , $F$ and $Q$ are in a line on level ground and $P Q = 85$ m. Find the height of the tower $F T$ .
[4 marks]
Not drawn accurately
Answer

Practice

A-Level

C1
In triangle $A B C$ , $A B = x$ cm, $A C = (x + 5)$ cm and angle $B A C = 3 0^{\circ}$ . The area of the triangle is $9$ cm². Find the value of $x$ .
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$

C2
In triangle $A B C$ , $A B = 10.5$ cm, angle $C A B = 4 8^{\circ}$ and angle $A B C = 7 1^{\circ}$ . Find the area of the triangle, to 3 s.f.
[4 marks]
Not drawn accurately
Answer
C3
In triangle $P QR$ , $P Q = 7$ cm, $QR = 8$ cm and $P R = 13$ cm. Show that angle $P QR = 12 0^{\circ}$ , and find the exact area of the triangle.
[4 marks]
Not drawn accurately
Answerarea $= \dots\dots$

Exam-style questions

Edexcel exam style

D1
A garden $A B C D$ has four straight fences. $B$ is $40$ m due north of $A$ . $C$ is $55$ m from $B$ on a bearing of $10 4^{\circ}$ . $C D = 48$ m and angle $A C D = 5 8^{\circ}$ .
Not drawn accurately
(a)
Find the length $A C$ , to 3 s.f.
[3 marks]
Answer
(b)
Hence find the area of the garden, to 3 s.f.
[4 marks]
Answer
Total for D1: 7 marks

D2
In triangle $P QR$ , $P Q = 14$ cm, $QR = 11$ cm and angle $QR P = 4 0^{\circ}$ . Sam writes: area $= \frac{1}{2} \times 14 \times 11 \times sin 4 0^{\circ} = 49.5$ cm².
Sam's answer is wrong.
Not drawn accurately
(a)
Explain the mistake Sam has made.
[2 marks]
(b)
Find angle $QP R$ . Explain why it has only one possible value.
[3 marks]
Answerangle $QP R = \dots\dots$
(c)
Hence find the correct area of the triangle, to 3 s.f.
[3 marks]
Answer
Total for D2: 8 marks
D3
In triangle $P QR$ , $P Q = (x + 3)$ cm, $P R = (9 - x)$ cm and angle $QP R = 15 0^{\circ}$ , where $- 3 < x < 9$ . The area of the triangle is $A$ cm².
Not drawn accurately
(a)
Show that $A = \frac{1}{4} (27 + 6 x - x^{2})$ .
[3 marks]
(b)
Hence find the maximum value of $A$ , and the value of $x$ at which it happens.
[3 marks]
Answer $A = \dots\dots$ $x = \dots\dots$
(c)
Given that $A = 8$ , find the two possible values of $x$ .
[3 marks]
Answer $x = \dots\dots$ or $\dots\dots$
Total for D3: 9 marks
D4
Triangle $A B C$ has perimeter $30$ cm. $A B = 12$ cm and angle $B A C = 12 0^{\circ}$ .
Not drawn accurately
(a)
Show that $A C = 3.75$ cm.
[4 marks]
(b)
Hence find the exact area of the triangle.
[2 marks]
Answer
(c)
Find the shortest distance from $A$ to the line $B C$ , to 3 s.f.
[2 marks]
Answer
Total for D4: 8 marks

Extension

Stretch

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

E1
A triangle has sides $10$ cm, $17$ cm and $21$ cm. A circle is drawn inside the triangle, touching all three sides. Find the exact radius of the circle.
[6 marks]
Hint 1 · What to try
Join the centre of the circle to the three corners. This splits the triangle into three triangles, each with height equal to the radius.
Hint 2 · The first line
Area $= \frac{1}{2} r (10 + 17 + 21) = 24 r$ . Now find the area another way: the cosine rule for one angle, then $\frac{1}{2} ab sin C$ .
Hint 3 · The full method
$cos C = \frac{1 0 ^{2} + 1 7 ^{2} - 2 1 ^{2}}{2 \times 10 \times 17} = - \frac{13}{85}$ , so $sin C = \frac{84}{85}$ and the area is $\frac{1}{2} \times 10 \times 17 \times \frac{84}{85} = 84$ . So $24 r = 84$ and $r = 3.5$
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the area of a triangle using half ab sin C.
I can find an angle or a side when I know the area.
I can choose the sine rule, the cosine rule or the area formula.
I can solve problems with heights, bearings and quadrilaterals.

Answers and mark scheme

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