Section T Similar and congruent shapes
Two shapes are similar if one is an enlargement of the other (even if it is in a different position and
orientation). There is a constant scale factor of enlargement from one to the other so the ratio of a
side on one of the shapes to the corresponding side on the other shape is always the same.
Two shapes are congruent if they are similar and the same size, even if they are in different
positions and orientations. Remember that if one shape is a reflection of another then they are
congruent.
Triangles are particularly important in these situations. If two triangles, P and Q, are similar then
each angle in P is the same as each angle in Q. We describe this condition as AAA, meaning that
there are three pairs of equal angles. With congruent triangles we are also interested in pairs
of equal length sides, which are denoted by the letter S. The conditions for two triangles to be
congruent are:
SSS – each side in one triangle matches a side in the other
ASA – two angles and the enclosed side in one triangle match two angles and the enclosed side
in the other
SAS – two sides and the enclosed angle in one triangle match two sides and the enclosed angle
in the other
The order of letters here is important – ASS is not a congruence condition, as there is sometimes
more than one triangle that can be drawn from this information. There is a common exception to
this – if the angle is a right angle then we can use a fourth congruence condition:
RHS – in right-angled triangles, the hypotenuse and one other side in one triangle match the
hypotenuse and one other side in the other triangle
Worked example T.1
Triangles ABC and XYZ are similar. Find the length of the side XY.
A
X
2
5
C
6
8
First find the scale factor of
enlargement by comparing
equivalent sides
Apply the scale factor to find XY
Z
Y
B
Scale factor =
So XY =
Cambridge Mathematics for the IB Diploma Higher Level © Cambridge University Press, 2012
XZ 2
=
AC 5
2
2
× AB = × 6 = 2.4
5
5
Prior learning T: Similar and congruent shapes 1
Exercise T
1. In the following diagrams triangle ABC is similar to triangle XYZ, with angle A = X and
B =Y . Find the length marked p.
(a)
A
5
8
X
2
C
(b)
B
p
Z
Y
A
Z
5
3
Y
p
2
X
4
C
(c)
B
A
X
p
2
6
5
C
B
4
Z
Y
2. State whether information given is sufficient to show that the pairs of triangles below are
congruent. If they are, state the congruence condition.
(a)
4
4
4
2
4
Cambridge Mathematics for the IB Diploma Higher Level © Cambridge University Press, 2012
2
Prior learning T: Similar and congruent shapes 2
(b)
7
3
40◦
40◦
7
(c)
3
2
2
5
5
10◦
10◦
(d)
40◦
6
30◦
40◦
Cambridge Mathematics for the IB Diploma Higher Level © Cambridge University Press, 2012
6
30◦
Prior learning T: Similar and congruent shapes 3