At least with triangles
we are back to straight edges. This means that there is nothing
tricky about the perimeter and area formulas. We will, however,
need to spend some time after that to review angles, as well
as the properties of some special triangles.
The perimeter of a triangle
is the sum of the lengths of the sides.
Perimeter
= a + b + c
Cool fact: Any side of a triangle is always shorter than
the sum of the other two sides. This is called the triangle
inequality theorem. Think about it for a second.
Imagine you have a ten foot piece of wood, and two four foot
pieces of wood. Could you connect them at their ends to make
a triangle? No, it just wouldn't work. The four foot sections
wouldn't be long enough. And you could have known this before
trying, because 10 > 4 + 4.
Triangle Inequality: For any triangle with
sides a, b, c : a < b + c
The area of a triangle is one half the
length of the base times the height.
Area = ½
Base × Height
Students often ask us where that formula comes from. The
shaded region above is a rectangle with Height = h and Width
= Base.
The area of that rectangle is Base × Height.
The triangle occupies exactly one half of the space of that
rectangle, and so the area of the triangle is half the area
of the rectangle.
Okay, that about does it for perimeter
and area. Let's move on to some more facts about triangles.
The vertices of a triangle (or points where the lines meet)
form angles. The sum of the measures of the angles within
a triangle is 180 degrees.
Interesting fact: If side x of a triangle is opposite a larger
angle than side y, then side x is longer than side y. Try
drawing some triangles and check this out.
Special Triangles
There are some special triangles to consider. If a triangle
has two = sides, it is called an isosceles triangle.
The angles opposite the two equal sides are equal (this follows
from the interesting fact mentioned on the last page).
If
a triangle has three equal sides, it is an equilateral triangle,
and all the angles measure 60 degrees.
Right Triangles
The most famous family of triangles are the right triangles.
Because of the right angle, it is easy to find the area of a
right triangle. Just turn it so that the right angle forms the
base and the height, and then apply the formula Area
= ½ b × h.
The Pythagorean theorem
helps you solve for the third side of a right triangle when
you know two of the other sides. The theorem says that
a2 + b2 = c2
The 3-4-5 Right Triangle
And within the family of right-angled triangles, there are some
even more special triangles.
One of the nicest is the 3-4-5
triangle.
52 = 32
+ 42
25 = 9 + 16
There are also the multiples of (3,4,5). That is, right triangles
can come in the proportions (6,8,10) and ( 12,16,20) , etc.
Always study a question with a right angle carefully to see
if it contains a 3-4-5 triangle, or a triangle derived from
one (i.e. one whose dimensions are scaled).
There are two other special right triangles.
One is the isosceles right triangle, where the other two angles
measures both equal 45 degrees. Here both these two angles and
the corresponding opposite sides are =.
In this triangle, the length of the hypotenuse equals the
length of either leg times the square root of 2.
Finally there is the 30-60-90 triangle, where the sides can
also be expressed as convenient proportions.
Keep an eye out for these triangles appearing either on their
own, or as parts of more complicated geometrical figures.
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