Sec 2 · G1 (N(T))

Congruence and similarity

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

1. 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

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2. 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 →

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1. 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 →

2. 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 →