Download 317 Chapter 44: Similar Triangles Ratios and

Chapter 44: Similar Triangles
Ratios and angles come together in the area of similar triangles. Similar triangles are key
to understanding trigonometry, but are also valuable in solving a number of problems.
Two geometric objects are called congruent if they have the exact same shape and size.
This means that if we slide one of them around (a movement or transformation called a
translation), rotate it, and/or take its mirror image (called a reflection), then we can get the first
object to exactly line up with the second one. This is easiest to do with shapes cut out of paper –
if one shape can be placed on top of the other so that they match up exactly, then they are called
congruent. Having children move shapes around like this is an excellent way to prepare them for
the advanced concepts of functions and groups. This is also a nice way to emphasize the
geometric concept of congruence. Basically, you put a number of shapes on a table, then ask the
kids to identify the ones that can be matched up exactly.
For example, in the figures below, shapes a, b, c, and d are all congruent, but e is not.
We could line up a and b by sliding one of them horizontally with a translation. We can rotate a
counter-clockwise 90° and then translate it over to line up with c. We will need to flip a over
with a reflection to match it up with d.
All of these tranformations preserve the size and shape of the original object. There is
another transformation, called a dihilation, which proportionally shrinks or enlarges shapes. For
example, if we double the height and width of the shape a above, it will look like f below:
The quadrilaterals a and f are called similar. Two figures are similar if they have the
same shape, but not necessarily the same size. When shapes are similar, we can talk about their
317
corresponding parts. If we were to shrink or enlarge one of the shapes and then move it to line
up with the other, the corresponding parts are the ones which would line up. For example, the
acute angles at the tops of a and f correspond and the sides between their right angles
correspond:
No matter how we might shrink or enlarge quadrilateral a, for example, the left-hand side
will always be the longest. Similarly, the left-hand side of quadrilateral f will always be the
longest. Thus, since these two shapes are similar, their left-hand sides must correspond.
The amount by which one shape must be shrunk or enlarged to become the size of a
similar shape is called the scale factor. In the above example, the scale factor from a to f is 2
because each side was doubled in size to make shape f. To go from f to a, on the other hand, we
would need a scale factor of 12 in order to make each side half as long.
When a shape is scaled up or down with a dihilation, its size can change considerably.
The angles, however, do not change. Thus, the corresponding angles of similar shapes are
congruent.
With triangles, there is a very simple test for similarity. If two triangles have the same
angles, then they are similar. For example, the two triangles below have the same three angles,
and thus are similar.
We have to take care to line up the corresponding sides correctly. The angles with the
same measure correspond, thus the sides between them correspond. For example, in the above
figures, the sides between the 35° and 50° correspond (and happen to be the longest sides of each
triangle). Similarly, the right side of the first triangle and the left side of the second correspond,
because these sides are between the corresponding 95° and 50° angles.
318
Once we know that two triangles are similar, it is only a matter of calculating the scale
factor. For example, look at the following triangles:
We know that the angles of a triangle add up to 180°. The first triangle has angles of 37°,
90°, and 180° - 37° - 90° = 53°. The second triangle has the same three angles (for the same
reason), and is thus similar to the first. Because the angles of any triangle add up to 180°, if
triangles have two angles in common like this, then they are similar.
Next, we see that the side of length 6 in the first triangle corresponds with the side of
length 15 in the second. We can use this information to calculate the scale factor which is used
to multiply the sides of the first triangle to get those of the second. We start with 6 × ? = 15 and
then work with the corresponding division problem: 15 ÷ 6 = ? A quick long division will
conclude that the scale factor is 2.5. This means that the other two sides of the larger triangle
can be found by multiplying 2.5 with the lengths of the corresponding sides:
With a pair of similar triangles, the lengths of corresponding sides all form the same
ratio. Using the above example, we have 25 : 10 is equivalent to 15 : 6 and 20 : 8. When these
are written as fractions, they will all reduce to the scale factor of 2.5.
Furthermore, any two sides of one triangle will form the same ratio as the corresponding
sides of the second triangle. If we take the height and base of the first triangle above, for
example, we get the ratio 6 : 8. The corresponding height and base of the second triangle form
an equivalent ratio of 15 : 20. Similarly, the ratio between the height and the longest side of the
two triangles are 6 : 10 and 15 : 25, which are equivalent ratios.
Any of these ratios can be used to calculate lengths on similar triangles.
319
A classic similar-triangles problem runs like this: "A 15-foot street light casts a 8-foot
shadow from a 5-foot tall woman. How far from the light is she standing?" It helps to draw the
following picture of the situation:
If we suppose that both the woman and the light are standing straight up and down at 90°
angles, then we can break the picture up into two similar triangles. This is because both triangles
have a 90° angle and whatever angle is down at the tip of the shadow:
The side of length 5 on the smaller triangle corresponds with the length of 15 on the
bigger triangle, and thus the scale factor must be 15 ÷ 5 = 3. We can find the base of the larger
triangle by multiplying 8 × 3 = 24. If we go back to the original picture, we can thus conclude
that the woman is standing 24 – 8 = 16 feet away from the street light:
320
You can have your students figure out the properties of similar triangles with protractors,
rulers, and paper. Break the class up into four groups. Have one group draw a 10-centimeter
line across the bottom of a page. Another group can draw a 15-centimeter line on their papers.
The other two groups can draw 20 and 25-centimeter
lines. Next, each student draws a 30° angle from one
end of the line and a 52° angle from the other. These are
then extended until they connect, forming a triangle. As
in the last chapter, you can quiz your students to see if
they can guess the measure of the third angle (180° - 30°
- 52° = 98°).
By doing this, all of the triangles will be similar, but of four different sizes. Next, have
each student measure the other two lengths of the triangle (to the nearest half-centimeter), then
report all these lengths up to the front of the room. Have each group report the lengths of their
(ideally) identical triangles as a team, to help eliminate measuring errors. The measurements
ought to come out roughly as follows:
longest side
medium side
shortest side
10 cm
8 cm
5 cm
15 cm
12 cm
7.5 cm
20 cm
16 cm
10 cm
25 cm
20 cm
12.5 cm
You can then discuss the data as a class, looking for patterns. Hopefully someone notices
that all three sides of the 20 cm triangle are double those of the 10 cm triangle. If someone
hypothesizes that a new triangle, drawn with a 5cm side and the same 30° and 52° angles, will
have exactly half the lengths, then let them make one and try. Similarly, find some large paper
for anyone who wants to try to start with a 30 cm line. Ideally, the patterns will gradually be
discovered. We can either divide one side by another for each triangle and get the same ratio
(for example, the shortest side of each is half the length of the largest). We could also compare
the lengths of one side to another. For example, we could compare the longest and shortest sides
of the smallest two triangles. The longest side of the bigger triangle is 15 cm, which is 1.5 times
longer than the 10 cm longest side of the smallest triangle. Similarly, the shortest side of the
larger triangle, 7.5 cm, is also 1.5 times the size of the 5 cm smallest triangle.
This, of course, leads to a discussion of similar triangles. From here, you can point out
all the ratios and proportions that your students have overlooked, and go on to explain the
definition, properties, and uses of similar triangles.
321
One interesting aspect of similar triangles we have not yet covered is how the area of
similar triangles relate. If our scale factor is 3, for example, then both the base and height will be
multiplied by 3, as in the example shown below:
The triangle on the left has an area of 12 u 4 u 5 = 10 in2. The triangle on the right has an
area of 12 u 12 u 15 = 90 in2. This area is 9 times bigger. It is no coincidence that 9 = 3 × 3,
because our scale factor increased the area of the triangle in two different dimensions.
It is worth computing the areas of some similar triangles, to see if your students can
detect this pattern. It helps to use right triangles because then the height can be one of the sides:
In this example, the scale factor is 4, which means that the area of the second triangle is 4
× 4 = 16 times bigger.
A fun and educational exercise for a class of children is to teach them the grid method for
copying a picture or photograph. There are differing levels of set-up which can be put into this
project. For the most extensive (and fun), each child should be encouraged to bring in a 3 × 5
photograph that they would like to copy. To avoid all the hassles of lost and forgotten photos,
you could provide photos instead or in addition to these and let the class choose them. For a less
fun (but far easier) project, you can photocopy all of the necessary materials and skip all student
choice and selection. Instead of photos, you can have students pick images from magazines.
However, it will help greatly if all the images used are the same size.
322
You should prepare in advance with a number of transparency sheets that are 3 × 5 in
dimension and have lines drawn horizontally and vertically every half inch. If you carefully
draw four of these onto a sheet of paper, you can photocopy it onto transparency sheets and then
cut them out. To be extra clever, you can tape the top and bottom to a piece of oak tag the same
size. Each student should get one of these and slide their photograph in behind the sheet:
Each student should also be provided with a piece of paper with a 6-inch by 10-inch grid,
divided up with lines every 1-inch. These can also be photocopied easily onto paper. Of course,
without a copier, you can have your students measure half-inch marks on their photos and draw
lines, then draw 1-inch marks on a piece of paper and draw lines, but this could take a very long
time.
At this point, have your students copy the photograph onto the piece of paper. They
begin with the square in the upper-left corner of the photo and copy just that square into the
corresponding square on the paper. The key is to look for lines in each square, keeping track of
where the line crosses the edge of the square. For example, consider the following situation:
The top row has just about nothing to draw, except
maybe some lines along the very bottom of the second and
third squares. The first square of the second row, however, has
a slightly-curved line that runs from the middle of the bottom
up to about a third-of-the-way down from the right-hand edge. Thus, we draw a line as similar
as possible in the first square of the second row from the top of the big piece of paper.
323
In this fashion, the task of copying a picture can be broken down into copying a large
number of small, simple squares. Not only does this make a complicated job easier, but it also
guarantees to preserve the shape, perspective, and proportions of the original image. It will take
some time and patience to complete the picture, but artwork always takes time and patience. If
your students want to complete the picture in the style of the great artists, they should then color
in the picture (and thereby try to cover up the grid lines).
Hopefully, your students will be able to appreciate the fact that this exercise demonstrates
the scaling and proportion concepts which are key to the idea of similarity.
Questions:
(1) Find the length of the side marked x in the following figure:
(2) Find all the missing lengths of the following triangles. Next, name all the sets of equivalent
ratios which can be obtained from the fact that these triangles are similar.
(3) Two triangles are similar. One triangle has 25 times the area of the first triangle. The base of
the small triangle is 9 cm long. What is the length of the corresponding base of the big triangle?
324