There are 18 basic types of questions you can
solve with the Pythagorean Theorem and trigonometry.
The first 3 types in the "Practice" section in the
interactive activity below are devoted to Pythagoras's Theorem.
If you know 2 of the three sides of a right
triangle use the Pythagorean Theorem to calculate the third
side.
The last 15 types involve trigonometry functions.
These types are also organized into groups of three similar
functions. After completing two of these you will probably
know what to do in the third one before looking at the examples.
The "Introduction" reviews the
Pythagorean theorem. "Naming Sides" reviews the
trigonometry triangle names required to use "Soh Cah
Toa". The "Practice" section has 5 sections
of problem types.
Problem Types 1-3: Pythagorean Theorem
Problem Types 4-6: Calculate the sine,
cosine and tangent ratios given one reference angle.
Problem Types 7-9: Calculate the sine,
cosine and tangent ratios given two appropriate sides.
Problem Types 10-12: Calculate the reference
angle given two appropriate sides using trigonometry functions.
Problem Types 13-18: Calculate the length
of one of the sides given appropriate side and reference
angle using trigonometry functions.
We can use trigonometryfunctions
to determine:
one acute angle if two sides are known
one of the sides if an acute angle and
one side are known
Examine the following 3-dimensional object. Vanishing
points are used to show perspective. Dragging the 3 dots will
change the vanishing points for length width and height.
The volume of a rectangular prism is a 3-dimensional
product (V=lwh or V=Bh), while surface area is a 2-dimensional
product (SA=2B + Ph).
Similar triangles have
the same shape, but the size may be different.
Remember "@" means "is
congruent to" and "~" is "similar to".
Examples
Corresponding Triangles
Corresponding Congruent
Angles
Corresponding Proportional Sides
a/f = b/d = c/e = factor
DABC ~ DFDE
<A @ <F
<B @ <D
<C @ <E
a/f = 6/3 = 2
b/d = 8/4 = 2
c/e = 10/5 = 3
DABC @ DFDE
<A @ <F
<B @ <D
<C @ <E
a/f = 3/3 = 1
b/d = 4/4 = 1
c/e = 5/5 = 1
Two triangles are similar if:
two pairs of corresponding angles are congruent (therefore
the third pair of corresponding angles are also congruent).
the three pairs of corresponding sides are proportional.
Notice the corresponding angles for the two triangles
in the applet are the same. The corresponding sides lengths
are the same only when the scale factor slider is set at
1.0. Study the side lengths closely and you will find that
the corresponding sides are proportional.
If you know triangles are similar, you
can use the proportion of corresponding sides to help determine
an unknown dimension.
Study the object below. You can change
the triangles by dragging on the slider or dragging vertex
A or B. The proportions of the corresponding pairs of
sides changes as the scale slider position changes. Each
time you stop dragging, look at the proportions. If one
of the sides was unknown you could use two pairs of corresponding
sides to calculate the missing dimension.
Congruent triangles are
a special type of similar triangles. Congruent triangles have
the same shape (similar triangles) and size.
Increase/decrease <A and <B
by clicking and dragging the vertices of the left triangle below.
Notice the corresponding angles (<D and
<E) remain congruent. Since the sum of three
angles must be 180o, the third pair of corresponding
angles must also be congruent when the first two pairs of corresponding
angles are congruent.
In the applet above:
<A @ <D, <B @ <E,
<C @ <F
Notice the size and shape of the new pairs
of triangles remains the same. The patterns you
may have observed and need to know for congruent triangles are
displayed below. Remember "@"
means "is congruent to".
Two triangles are congruent if:
all 6 pairs of corresponding angles and sides are congruent.
Corresponding Triangles
Corresponding Congruent
Angles
Corresponding
Congruent Sides
factor = 1
a/f = b/d = c/e = 1
DABC
@ DDEF
<A @
<D
<B @ <E
<C @ <F
AB @
DE
BC @ EF
AC @ DF
The factor for congruent triangles
is 1. Remember this fact for future problem solving activities.
If all 3 pairs of corresponding sides in two triangles
are the same, then the triangles are congruent.
Move the corners A, B or C of the triangle
above. Watch how DDEF
changes. Since the 3 pairs of corresponding sides of the
two triangles are equal each time you manipulate the applet,
the SSS (Side, Side, Side) Congruence Relation
proves each pair of triangles are congruent.
If three sides of one triangle
are congruent to three sides of a second triangle, the two
triangles are congruent.
ASA
If two angles
and the included side of one
triangle are congruent to two angles and the included side
of another triangle, the triangles are congruent.
SAS
If two sides and the
included angle are congruent to two sides and the included
angle of a second triangle, the two triangles are congruent.
AAS
If two angles and a non-included
side of one triangle are congruent to two angles and the corresponding
non included side of another triangle, the two triangles are
congruent.
Hyp-S
If the hypotenuse and the
leg of one right triangle are congruent to the corresponding
parts of the second right triangle, the two triangles are
congruent
Two triangles with two
sides and a non-included angle equal may or may not be congruent.
AAA
If two angles
on one triangle are equal, respectively, to two angles on
another triangle, then the triangles are similar, but not
necessarily congruent.
dilatation - a transformation that changes the size of
an object
enlargement - a dilatation where the image is larger than
the original object
reduction- a dilation where the image is smaller than
the original object
scale factor can be calculated from the ratio:
length from origin to image
--------------------------------------
length from origin to original
dilatation centre - lines drawn through corresponding
image vertices will meet at the dilatation centre
coordinates - an ordered pair of the form (x, y) that
locates a point on a coordinate plane
corresponding sides - sides that have the same relative
positions in geometric figures
proportional - two objects are proportional if the all
the ratios of the corresponding sides are the same
perspective - the different views of an object - top,
bottom, side, front
vanishing point - is the point at which, if two parallel
lines, or walls were extended into the distance as far as you
could see, would look like they meet or vanish into the distance.
A great example of this is railway tracks. When you look down
the tracks, it looks like the tracks eventually come together.
There are 18 basic types of questions you can
solve with the Pythagorean Theorem and trigonometry.
The first 3 types in the "Practice" section in the
interactive activity below are devoted to Pythagoras's Theorem.
If you know 2 of the three sides of a right
triangle use the Pythagorean Theorem to calculate the third
side.
