To understand the principle of similarity in triangles
[LO 1.2, 1.4, 3.5]
In a previous exercise you were asked to draw ΔBAT with B=48°, T=65° and A=67°. As you noticed, it was possible to draw many triangles according to these specifications, but they were not congruent. This is because nothing specified the size of the triangle, so some were bigger and some were smaller.
Work in groups of four or five. Measure the sides of your triangle. Each uses his own Δ BAT to complete his row in the table below. Where you divide, give your answer rounded to
one decimal place.
Learner
AB
AT
BT
BT AT
AB AT
BT AB
In the next exercise you must draw two isosceles triangles with angles 80°, 50° and 50°. Make the one triangle about two or three times as big as the other. Work very accurately.
Call the small triangle DEF (F = 80°) and the big one OPT (T = 80°). Measure all the sides and complete the table from your measurements.
OP
PT
OT
DE
EF
DF
OP DE
PT EF
OT DF
Assignment:
Study the two tables (especially the last three columns of both tables). What do you notice?
Write a very clear explanation of why these calculations work out the way they do.
These triangles are not congruent, as their sizes are different, even if their angles agree. When two triangles have equal angles but different sizes, we call them
simil a
r .
The sign is , so that ΔDEF ΔOPT from the last table.
All triangles with three angles equal are automatically
simil a
r . If they have the same size, then they are
congruent as well.
Similar triangles have sides that are in the same proportion. This is what we see from the two tables we completed. The fractions that were calculated from the side lengths give us the ratios between sides.
From the first table we see that the ratios between the sides of similar triangles are the same. From the second table we see that the ratios between corresponding sides of two similar triangles are the same. This ratio is called the
proportion a
l const a
nt for the two triangles.
We know two things from the facts we have learned about similar triangles:
Firstly, if we have two triangles with equal angles (
equi a
ngul a
r tri a
ngles ), then we know they are similar and therefore the sides must be in proportion.
Secondly, when we have triangles with sides in proportion, we know they must have equal angles because they must be similar.
Example:
What can you say about the two triangles below? Calculate the values of
x and
y .
To find out whether the triangles are similar, we must find either equal angles or sides in proportion. In this problem, we can say the angles are equal but we cannot say the sides are in proportion. In other problems, it may be the other way around. .
We set it out this way:
1. A = 65° because the angles of ΔABC add up to 180°
F = 35° because the angles of ΔDEF add up to 180°
2. The triangles have equal angles, therefore they are similar, so
ΔABC ΔDEF (equal angles)
3. This means that the sides must be in proportion.
4. Find out what the proportional constant is. AC = 16 and DF = 8.
