TOPIC 6: SIMILARITY ~ MATHEMATICS FORM 2
Two geometrical figures are called similar if they both have the same shape. More precisely one can be obtained from the other by uniformly scaling (enlarging or shrinking). Possibly with additional translation, rotation and reflection. Below are similar figures, the figures have equal angle measures and proportional length of the sides:
Angle
A corresponds to angle A’, angle B corresponds to B’, angle C corresponds to C’ also each pair of these corresponding sides bears the same ratio, that is:
Since all sides have the same ratio i.e. they are proportional and the corresponding angles are equal i.e. angle A = angle A’, angle B = angle B’ and angle C = angle C’, then the two figures are similar. The symbol for similarity is ‘∼‘
Similar Polygons
Triangles are similar if the only difference is size (and possibly the
need to Turn or Flip one around). The Triangles below are similar
- All their angles equal
- Corresponding sides have the same ratio
Since
we know that, similar triangles have equal ratio of corresponding
sides, finding the ratio of the given corresponding sides first thing:
How to find if Triangles are Similar
Two Triangles are similar if:
- All their equals are angles
- The corresponding sides are in the same ratio
But we don’t need to know all three angles and all three sides, even two or three are enough.
The theory is also called Side-Splitter theorem. Let ABC be any Triangle and DE is drawn parallel to BC, then AD/DB = AE/EC.
The Triangles ADE and BD have exactly equal angles and so they are similar (recall that the two Triangles are similar by AA).
There are three ways to find that the two Triangles are Similar
If two of their Angles are equal then the third Angle must also be equal, because Angles of a Triangle add up to1800. In our case, our third Angle will be:
Therefore, AA can also be called AAA because when two angles are equal then all three Angles must be equal.
- The ratio between two sides is the same as the ratio between the other two sides
- The included Angles are equal
From our example, we see that, the side AB corresponds to side XZ and side BC corresponds to side YZ, thus the ratios will be:
The information is enough to tell us that the Triangles are Similar.
a:x = 6:7.5 = 12:15 = 4:5
TOPIC 6: SIMILARITY ~ MATHEMATICS FORM 2
Exercise 1
2. ABC is a Triangle in which AC is produced to E and AB is Produced to D such that DE//BC. Show that AD:AB = DE:CB
Given Triangles ABC and PQR which are similar. If the lengths of sides
AC = 4.8cm, AB = 4cm and PQ = 9cm find the length of side PR if side AB
corresponds to PQ and BC corresponds to QR.
4. Prove that any two equilateral Triangles are similar.
