page 3 of 7
1.
Angles and Lines
2. Intersecting
Angles
3. Triangles
4. Circles
5. Perimeters
& Areas
6. Solids
7. Coordinate
Geometry
Triangles
A
triangle has three sides and three angles; the sum of its three
angles is 180
. There are three triangles that
are particularly important to us: isosceles, equilateral
and right. An isosceles triangle has two equal
sides; the angles opposite the equal sides are also equal. All
three sides of an equilateral triangle are equal; each
of its three angles are 60. A
right triangle is a triangle that has a 90
angle; the Pythagorean Theorem states that c
= a + b , where c
is its hypotenuse and a and b are its legs. The
hypotenuse is always opposite the 90angle
and the legs are always shorter than the hypotenuse.
There are certain right triangles
that show up often on the test. The 3 - 4 - 5 triangle may be
the most popular; note that the Pythagorean Theorem is satisfied
since 5 = 3 + 4. The 5-12-13 triangle also surfaces occasionally.
The second most popular triangle is the 30- 60- 90 triangle because the ratio of its short
leg to its hypotenuse is 1 : 2. The 45- 45- 90 triangle has equal legs and is also encountered
quite often. In the case of the 45- 45- 90 triangle, the length of each leg
is 
times the hypotenuse; in
other words, the hypotenuse is 
times
one of the legs.
Commonly Used
Triangles
3 - 4 - 5 triangle |
30- 60- 90 triangle |
5-12-13 triangle |
45- 45- 90 triangle |
A
triangle may not have the actual dimensions shown above, but
may have a multiple of the dimensions. For example, if two legs
of a right triangle have dimensions of 9 and 12, their ratio
is 3:4, so the triangle is a 3 - 4 - 5 triangle but three times
larger than the base triangle; the hypotenuse is 3 X 5 = 15.
The hypotenuse of a 45- 45- 90triangle has dimensions 
larger than the legs, as shown below.
If the hypotenuse of a 30- 60- 90triangle has dimensions as shown below,
the side opposite the 30
angle is 1/2 the length of
the hypotenuse. The side opposite the 60 angle
is 
/ 2, multiplied by the hypotenuse.
800score.com Strategy: Always have memorized, it is 1.732 and the is 1.414. Why memorize these? Because when you are doing geometry questions you will often have to guess answers within rough paramerters. This may allow you to better guesstimate your answers. So lets say you know the answer to a given question is probably between 25 and 30, but you get 20. Now you can sub for 1.4 and get 28 (which fits your guesstimate).
Example 1
For the triangle shown, find L.
Solution
The small box in the corner signifies a right triangle. The ratio
of the two legs is 12/16 = 3/4. It is a 3-4-5 triangle. It is
4 times the base 3-4-5 triangle; consequently, its hypotenuse
L is L = 4 x 5 = 20.
Or we could have used the Pythagorean
Theorem to obtain:
L= 12 + 16
L= 400
L = 20
Example 2
Calculate the length
L for the triangle shown.
Solution
This is a right triangle, a 45- 45- 90 triangle. The length of a leg of such
a triangle is times the hypotenuse.
This gives
Example 3
A given isosceles triangle has two equal angles of 30. The side common to the 30 angles
has a length of 4. How long are the equal sides?
Solution
A sketch of the triangle is always helpful. Let x be the unknown
length. You know how to use the properties of a right triangle
to solve for the sides of a triangle, so if you have to solve
for the side of a different kind of triangle, you can use a right
triangle within the given triangle. Can you see how one of the
triangles we've just discussed could be helpful in solving the
problem? By dividing the isosceles triangle into 2 right triangles,
we get two 30- 60- 90triangles.
The ratio of the side adjacent to the 30 angle
and the hypotenuse is 
. Hence,
Example 4
A triangle has angles of 45and 75 The
side opposite the 45angle has a length of 6. What is the
length of the side opposite the 75 angle?
Solution
Sketch the triangle. The remaining angle is 180 - (75 + 45) =
60. Again, see if you can solve the problem
by creating right triangles. Form two right triangles and label
the unknowns x, y, z. The side adjacent to the 60 angle is 1/2 the hypotenuse.
Hence, y = 3. The side opposite the 60 angle
is x = 3 (the triangle is 3 times as big
as the base 30-
60 - 90 triangle shown previously). Since the legs of
a 45- 45- 90 triangle are equal, z = x = 3. The length is then
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