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.

Isosceles Triangle A = C

Equilateral Triangle A = B = C = 60

Right Triangle c = a + b B + C = 90

      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 hypotonuse. The side opposite the 60 angle is / 2, multiplied by the hypotonuse.

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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