GCSE Maths · Higher · Chapter 12: Similarity · Lesson 1

Similar triangles

Grades 5–8About 50 minutesCalculator allowed38 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Share £48 in the ratio $1 : 7$ . Find the larger share.
[1 mark]
Answer
A2
From chapter 10
A taxi costs $C = 19 m + 31$ pence for $m$ miles. What does the 19 tell you?
[1 mark]
Answer
A3
Last lesson
From 19 m away the angle of elevation to the top of a tree is $4 5^{\circ}$ . How tall is the tree?
[1 mark]
Answer
A4
Key Stage 3
Write $\frac{7}{20}$ as a decimal.
[1 mark]
Answer

Grades 5–6

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

Example 1

Triangles $A B C$ and $P QR$ are similar, with $A$ , $B$ and $C$ matching $P$ , $Q$ and $R$ . Work out $QR$ .

Not drawn accurately

Hint Find the scale factor from a pair of matching sides that are both known.

B1
The two triangles are similar. Matching angles are marked the same. Work out the length $x$ .
[3 marks]
Not drawn accurately
Answer

Example 2

$D$ is on $A B$ and $E$ is on $A C$ , with $D E$ parallel to $B C$ . Work out $B C$ .

Not drawn accurately

Hint The triangles share angle

A

, and

D E ∥ B C

makes the other angles equal.

A D

matches

A B = 15

cm, not

D B

B2
$D$ is on $A B$ and $E$ is on $A C$ , with $D E$ parallel to $B C$ . Work out the length of $E C$ .
[3 marks]
Not drawn accurately
Answer

Grades 5–7

C1
In triangle $A B C$ , angle $A = 3 8^{\circ}$ and angle $B = 6 4^{\circ}$ . In triangle $P QR$ , angle $P = 6 4^{\circ}$ and angle $Q = 7 8^{\circ}$ . Show that the triangles are similar.
[2 marks]
Not drawn accurately
C2
$A B$ is parallel to $D E$ . The straight lines $A D$ and $B E$ cross at $X$ . Work out the length of $B E$ .
[3 marks]
Not drawn accurately
Answer

C3
A post and a tower stand on level ground. From $O$ , the tops of the post and the tower are in line. Work out the height $h$ of the tower.
[3 marks]
Not drawn accurately
Answer
C4
Are these two triangles similar? Give a reason.
[2 marks]
Not drawn accurately

Grades 6–8

D1
$D$ is on $A B$ and $E$ is on $A C$ , with $D E$ parallel to $B C$ . Mia says, "The scale factor is $10 \div 5 = 2$ , so $B C = 18$ cm."
Explain why Mia is wrong, and work out the correct length of $B C$ .
[3 marks]
Not drawn accurately

D2
Triangle $A B C$ has a right angle at $C$ . $A B = 20$ cm and $C D$ is perpendicular to $A B$ .
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(a)
Show that triangles $A C D$ and $A B C$ are similar, then work out $A D$ .
[4 marks]
Answer
(b)
Work out the perimeter of triangle $B C D$ .
[3 marks]
Answer
Total for D2: 7 marks
D3
Triangles $A B C$ and $P QR$ are similar. Work out $x$ . You must show your working.
[4 marks]
Not drawn accurately
Answer

Grade 8 and beyond

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

Colour one circle on each line.

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