Download Module 2 notes - The Pythagorean Theorem Right triangles contain

Module 2 notes
- The Pythagorean Theorem
Right triangles contain one right angle and two acute angles. In the
diagram below, the right angle is denoted by the square
symbol.
The hypotenuse is always the longest side in a right triangle, and it is
always across from the right angle.
The Pythagorean Theorem is a mathematical relationship between the
sides of a right triangle. It works only with right triangles!
Pythagorean Theorem: If the length of each leg is squared, their
sum will be equal to the square of the hypotenuse.The formula is given
as follows:
leg2 + leg2 = hypotenuse2
If you know the length of two sides, you can find the third side by
using this theorem.
Examples: Use Pythagorean Theorem to solve for the unknown
side in the two examples below.
If a = 3 and b = 4, find the length of c.
leg2 + leg2 = hyp2
32 + 42 = hyp2
9 + 16 = hyp2
25 = hyp2
5 = hyp
-Similar Triangles
Definition: Geometric figures are similar if they have the same shape
but not necessarily the same size.
For example, in the diagram below, the triangles ABP, ACQ, ADR, and
AES are similar because their shapes are the same although their sizes
are different. Triangles have the same shape when their
corresponding angles are the same.
Example these are similar shapes just different sizes…what makes
then similar is that they have the same angles.
-Trigonometric ratios
There are three trigonometric ratios used: sine, cosine, and tangent.
These ratios work with right triangles.
-The reference angle is one of the acute angles in the right triangle. It
can never be the right angle. The name of the reference angle is
included immediately after the ratio. In the definition above, the
reference angle is ( A). The reference angle may be represented by
any symbol, and one you will see often is the Greek letter theta (
).
The acronym SOH-CAH-TOA might help you remember these ratios.
Rules for Labelling Right Triangles
1. The leg across from the reference angle is always called the
opposite leg.
2. The hypotenuse is always fixed regardless of which angle is the
reference angle. It is always the longest side and is always across from
the right angle.
3. The remaining leg, which is next to the given angle but not the
hypotenuse, is called the adjacent leg.
Example 1: Find the unknown side in the following triangle.
To find a missing side or angle, always follow these steps:
Step 1: Draw a diagram and label it.
Draw a diagram and label the given information and the unknown. If it is a word problem,
be sure to draw a representative diagram.
Step 2: Select the appropriate trigonometric ratio.
Select the trigonometric ratio that contains the labelled sides in the diagram from Step 1.
Labelling what is given and what is unknown in Step 1 will ensure that you select the
correct trigonometric ratio.
Step 3: Substitute into ratio.
Substitute the correct values from the diagram into the formula.
Step 4: Solve.
To isolate the unknown in the above equation, first multiply both sides of the equation by
"x".
Example 2:
Find the value of angle
We use the cosine ratio because labelling identified the adjacent side and hypotenuse.
To find the measure of any angle using the trigonometric ratios, use the inverse function
of sine, tangent, or in this case, cosine. The inverse cosine is determined by pressing the
second function button on your calculator and then cosine key.
Word Problems
To solve a right triangle means to calculate all unknown angle
measures and side lengths. This can be done through trigonometry,
the Pythagorean Theorem, and the fact that the sum of the interior
angles of a triangle is 180 degrees.
To solve problems, begin by sketching a representative diagram of the
problem. In addition, place the given information correctly on the
triangle.
When solving right triangles, angles of elevation and angles of
depression may be used.
-An angle of elevation is formed by the horizontal and a line of
sight above the horizontal.
-An angle of depression is formed by the horizontal and the line of
sight below the horizontal.