Sec 2 · G3 (Express)

Congruence and similarity

Practise congruence and similarity with step-by-step help. Free account.

1. Congruent figures

Term	Meaning
congruent	same shape and same size
$\triangle ABC \equiv \triangle PQR$	$A \leftrightarrow P$ , $B \leftrightarrow Q$ , $C \leftrightarrow R$ , in that order
corresponding sides	equal: $AB = PQ$
corresponding angles	equal: $\angle A = \angle P$

📖 Worked example

In the diagram, triangle $ABC$ is congruent to triangle $RPQ$ . Find $x$ .

Step 1 of 2

Match the letters in order

\triangle ABC \equiv \triangle RPQ

, so

BC

matches

PQ

Practise this →

2. Similar figures

Triangle ABC with AB = 4 and triangle DEF with DE = 6; angle A equals angle D and angle B equals angle E, so the triangles are similar with scale factor 1.5.

Similar figures	Example: $\triangle ABC$ and $\triangle DEF$
corresponding angles equal	$\angle A = \angle D$ , $\angle B = \angle E$
corresponding sides in the same ratio	$\dfrac{DE}{AB} = \dfrac{EF}{BC} = \dfrac{DF}{AC}$
scale factor	$\dfrac{6}{4}$ , so $1.5$

AC = 3.2 cm: find DF

\displaystyle \dfrac{DF}{AC} = \dfrac{6}{4}

e.g.

\displaystyle \dfrac{DF}{3.2} = \dfrac{6}{4}

\displaystyle DF = 4.8

📖 Worked example

In the diagram, triangle $ABC$ is similar to triangle $QRP$ . Find $x$ .

Step 1 of 3

Match the sides by the order of the letters

BC

matches

RP

and

CA

matches

PQ

Practise this →

3. Perpendicular and angle bisectors

Perpendicular bisector of AB: arcs of the same radius from A and from B cross at P and Q; PQ meets AB at its midpoint M at right angles. X on PQ is the same distance from A and B.

Bisector of angle ABC: an arc about B cuts BA at X and BC at Y; equal arcs from X and Y meet at Z; BZ cuts the angle into two equal halves. Q on BZ has equal perpendicular distances QM and QN to the two arms.

Construction	Property	Example
perpendicular bisector of $AB$ : arcs of the same radius (more than half of $AB$ ) from $A$ and $B$ , join the two crossings	every point on it is the same distance from $A$ and $B$ ; it cuts $AB$ in half at $90^\circ$	$PA = 7$ cm, so $PB = 7$ cm
bisector of $\angle ABC$ : arc from $B$ , then equal arcs from $X$ and $Y$ , join $BZ$	every point on it is the same distance from the lines $BA$ and $BC$ ; it cuts $\angle ABC$ in half	$\angle ABC = 64^\circ$ , so $\angle ABZ = 32^\circ$
where two perpendicular bisectors of $\triangle ABC$ meet	the same distance from $A$ , $B$ and $C$	a mast the same distance from three towns

📖 Worked example

In the diagram, the perpendicular bisector of $AB$ has been constructed with compasses. It meets $AB$ at $M$ , and $P$ is a point on it. $AB = 8$ cm. Find the perimeter of triangle $PAB$ .

Step 1 of 2

Use the property of the perpendicular bisector

PB = PA

(

P

is on the perpendicular bisector of

AB

)

PB = 6

📖 Worked example

In the diagram, the bisector of $\angle ABC$ has been constructed with compasses, and $Q$ is a point on it. $QM$ is perpendicular to $BA$ and $QN$ is perpendicular to $BC$ . Find $x$ .

Step 1 of 2

BQ bisects angle ABC: the two halves are equal

\angle QBN = \angle ABQ

\angle QBN = 31^\circ

📖 Worked example

Three towns are at $A(1, 5)$ , $B(1, 1)$ and $C(7, 1)$ . A mast is to be built the same distance from all three towns. Find the coordinates of the mast.

Step 1 of 5

Points the same distance from B and C lie on

Same distance from

B

and

C

: on the perpendicular bisector of

BC

, the vertical line through its midpoint.

Practise this →

4. Congruent and similar polygons

Figure	Match	Use
$ABCD \equiv RSPQ$	letters in the order of the statement	matching sides and angles are equal
$ABCD \sim PQRS$	letters in order	scale factor $= \dfrac{PQ}{AB}$ ; angles equal
$\angle ADE = \angle ACB$ ( $\angle A$ shared)	$\triangle ADE \sim \triangle ACB$	$\dfrac{AD}{AC} = \dfrac{AE}{AB} = \dfrac{DE}{CB}$
$\angle ABD = \angle ACB$ ( $D$ on $AC$ )	$\triangle ABD \sim \triangle ACB$	$\dfrac{AB}{AC} = \dfrac{AD}{AB}$ , so $AB^2 = AD \times AC$
congruent triangles	matching angles equal	equal alternate (or corresponding) angles: the lines are parallel

Enlargement	Scale factor
on squared paper	$\dfrac{\text{length on the image}}{\text{matching length on the object}}$
$k < 1$	the image is smaller (a reduction)

📖 Worked example

On the squared paper, shape $B$ is an enlargement of shape $A$ . Find the scale factor of the enlargement.

Step 1 of 2

Count the squares along a pair of matching sides

Bottom side of shape

A

6

units; of shape

B

3

units

📖 Worked example

In triangle $ABC$ , $D$ lies on $AB$ and $E$ lies on $AC$ such that $\angle ADE = \angle ACB$ . Find $x$ .

Step 1 of 3

Name the similar triangles in matching order (∠A is shared)

\triangle ADE \sim \triangle ACB

(

\angle A

is common and

\angle ADE = \angle ACB

; AA)

Practise this →

5. Enlargement and scale drawings

Scale factor $k$	Image
$k > 1$	enlargement (bigger)
$0 < k < 1$	reduction (smaller)
image length	$k \times$ object length

Scale drawing 1 : 200, a wall 7.5 m long

\displaystyle 1 : 200 \text{, so } 1\text{ cm} : 2\text{ m}

e.g.

\displaystyle 7.5 \div 2

\displaystyle = 3.75

$3.75$ cm on the drawing

📖 Worked example

The plan of a house is drawn to a scale of $1 : 200$ . A room is $1.6$ cm long on the plan. Find the actual length, in metres, of the room.

Step 1 of 2

Write the scale as a ratio with units (1 m = 100 cm)

\mathstrut 1 : 200

\mathstrut 1\text{ cm} : 200\text{ cm}

1\text{ cm} : 2\text{ m}

Practise this →

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Sec 2 · G3 (Express)

Congruence and similarity

Practise congruence and similarity with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Congruent figures

Term	Meaning
congruent	same shape and same size
$\triangle ABC \equiv \triangle PQR$	$A \leftrightarrow P$ , $B \leftrightarrow Q$ , $C \leftrightarrow R$ , in that order
corresponding sides	equal: $AB = PQ$
corresponding angles	equal: $\angle A = \angle P$

📖 Worked example

In the diagram, triangle $ABC$ is congruent to triangle $RPQ$ . Find $x$ .

Step 1 of 2

Match the letters in order

\triangle ABC \equiv \triangle RPQ

, so

BC

matches

PQ

Practise this →

2. Similar figures

Similar figures	Example: $\triangle ABC$ and $\triangle DEF$
corresponding angles equal	$\angle A = \angle D$ , $\angle B = \angle E$
corresponding sides in the same ratio	$\dfrac{DE}{AB} = \dfrac{EF}{BC} = \dfrac{DF}{AC}$
scale factor	$\dfrac{6}{4}$ , so $1.5$

AC = 3.2 cm: find DF

\displaystyle \dfrac{DF}{AC} = \dfrac{6}{4}

e.g.

\displaystyle \dfrac{DF}{3.2} = \dfrac{6}{4}

\displaystyle DF = 4.8

📖 Worked example

In the diagram, triangle $ABC$ is similar to triangle $QRP$ . Find $x$ .

Step 1 of 3

Match the sides by the order of the letters

BC

matches

RP

and

CA

matches

PQ

Practise this →

3. Perpendicular and angle bisectors

Construction	Property	Example
perpendicular bisector of $AB$ : arcs of the same radius (more than half of $AB$ ) from $A$ and $B$ , join the two crossings	every point on it is the same distance from $A$ and $B$ ; it cuts $AB$ in half at $90^\circ$	$PA = 7$ cm, so $PB = 7$ cm
bisector of $\angle ABC$ : arc from $B$ , then equal arcs from $X$ and $Y$ , join $BZ$	every point on it is the same distance from the lines $BA$ and $BC$ ; it cuts $\angle ABC$ in half	$\angle ABC = 64^\circ$ , so $\angle ABZ = 32^\circ$
where two perpendicular bisectors of $\triangle ABC$ meet	the same distance from $A$ , $B$ and $C$	a mast the same distance from three towns

📖 Worked example

In the diagram, the perpendicular bisector of $AB$ has been constructed with compasses. It meets $AB$ at $M$ , and $P$ is a point on it. $AB = 8$ cm. Find the perimeter of triangle $PAB$ .

Step 1 of 2

Use the property of the perpendicular bisector

PB = PA

(

P

is on the perpendicular bisector of

AB

)

PB = 6

📖 Worked example

In the diagram, the bisector of $\angle ABC$ has been constructed with compasses, and $Q$ is a point on it. $QM$ is perpendicular to $BA$ and $QN$ is perpendicular to $BC$ . Find $x$ .

Step 1 of 2

BQ bisects angle ABC: the two halves are equal

\angle QBN = \angle ABQ

\angle QBN = 31^\circ

📖 Worked example

Three towns are at $A(1, 5)$ , $B(1, 1)$ and $C(7, 1)$ . A mast is to be built the same distance from all three towns. Find the coordinates of the mast.

Step 1 of 5

Points the same distance from B and C lie on

Same distance from

B

and

C

: on the perpendicular bisector of

BC

, the vertical line through its midpoint.

Practise this →

4. Congruent and similar polygons

Figure	Match	Use
$ABCD \equiv RSPQ$	letters in the order of the statement	matching sides and angles are equal
$ABCD \sim PQRS$	letters in order	scale factor $= \dfrac{PQ}{AB}$ ; angles equal
$\angle ADE = \angle ACB$ ( $\angle A$ shared)	$\triangle ADE \sim \triangle ACB$	$\dfrac{AD}{AC} = \dfrac{AE}{AB} = \dfrac{DE}{CB}$
$\angle ABD = \angle ACB$ ( $D$ on $AC$ )	$\triangle ABD \sim \triangle ACB$	$\dfrac{AB}{AC} = \dfrac{AD}{AB}$ , so $AB^2 = AD \times AC$
congruent triangles	matching angles equal	equal alternate (or corresponding) angles: the lines are parallel

Enlargement	Scale factor
on squared paper	$\dfrac{\text{length on the image}}{\text{matching length on the object}}$
$k < 1$	the image is smaller (a reduction)

📖 Worked example

On the squared paper, shape $B$ is an enlargement of shape $A$ . Find the scale factor of the enlargement.

Step 1 of 2

Count the squares along a pair of matching sides

Bottom side of shape

A

6

units; of shape

B

3

units

📖 Worked example

In triangle $ABC$ , $D$ lies on $AB$ and $E$ lies on $AC$ such that $\angle ADE = \angle ACB$ . Find $x$ .

Step 1 of 3

Name the similar triangles in matching order (∠A is shared)

\triangle ADE \sim \triangle ACB

(

\angle A

is common and

\angle ADE = \angle ACB

; AA)

Practise this →

5. Enlargement and scale drawings

Scale factor $k$	Image
$k > 1$	enlargement (bigger)
$0 < k < 1$	reduction (smaller)
image length	$k \times$ object length

Scale drawing 1 : 200, a wall 7.5 m long

\displaystyle 1 : 200 \text{, so } 1\text{ cm} : 2\text{ m}

e.g.

\displaystyle 7.5 \div 2

\displaystyle = 3.75

$3.75$ cm on the drawing

📖 Worked example

The plan of a house is drawn to a scale of $1 : 200$ . A room is $1.6$ cm long on the plan. Find the actual length, in metres, of the room.

Step 1 of 2

Write the scale as a ratio with units (1 m = 100 cm)

\mathstrut 1 : 200

\mathstrut 1\text{ cm} : 200\text{ cm}

1\text{ cm} : 2\text{ m}

Practise this →

Try a question: Congruence and similarity

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise congruence and similarity with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

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