Pythagorean Theorem
Unit #4: Trigonometry
[pic]
PYTHAGOREAN THEOREM
Investigate:
1. What type of triangle is in the centre of the picture?
2. Complete the table below
Side a = Area of A =
Side b = Area of B =
Side c = Area of C =
3. How is the area of A and the area B related to the area of C?
4. Name the famous mathematician who first figured this out.
RULE:
Pythagorean Theorem states …
There are two versions of the Pythagorean Theorem, and they depend on the hypotenuse.
Looking for the HYPOTENUSE…… Have the HYPOTENUSE ….
Example 1: Example 2:
a) Label the hypotenuse a) Label the hypotenuse
b) Calculate the missing side length b) Calculate the missing side length
Describe two ways to identify the hypotenuse in a right triangle.
(
(
[pic]
Justify your answer using your knowledge of the Pythagorean Theorem.
Pythagorean Theorem Practice
1. Solve for the missing side in the following triangles:
a) b)
2. A wall is supported by a brace 10 feet long, as shown in the diagram below. If one end of the brace is placed 6 feet from the base of the wall, how many feet up the wall does the brace reach?
[pic]
3. When Arnold swims laps in his rectangular swimming pool, he swims along the diagonal so he doesn’t have to turn around so often. Find the distance Arnold travels by swimming once along the diagonal.
4. The NuFone Communications Company must run a telephone line between two poles at opposite ends of a lake, as shown in the accompanying diagram. The length and width of the lake are 75 feet and 30 feet, respectively.
[pic]
What is the distance between the two poles, to the nearest foot?
5. A telephone pole support cable attaches to the pole 20 feet high. If the cable is 25 feet long, how far from the bottom of the pole does the cable attach to the ground?
[pic]
6. A baseball diamond is a square with sides 90 ft. What is the distance to the nearest tenth of a foot between 1st base and 3rd base?
7. In a computer catalog, a computer monitor is listed as being 19 inches. This distance is the diagonal distance across the screen. If the screen measures 10 inches in height, what is the actual width?
8. Joe has a 7m ladder. How far up the wall will the ladder reach if the foot of the ladder must be 2 m from the wall?
9. Given the following triangle
a. Find the length of the missing side
b. Find the perimeter of the triangle
c. Find the area of the triangle
5 cm
12 cm
10. Towns A, B, C, and d are situated as shown on the diagram.
A) How far is it from town B to
B town D?
B) How far is it from town B
10Km to town C?
A C
D
7Km 6Km
TRIGONOMETRY INTRODUCTION
Trigonometry is simply the study of __________________________________ measurements.
In grade 10, we will be dealing ONLY with __________________________ (90°) triangles.
Trigonometry looks at ______ of the three sides in relation to an "indicated" angle.
Before we can jump into trigonometry we need to know the proper names for the sides in a triangle
SIDES OF A TRIANGLE
Label the hypotenuse, the opposite, and the adjacent sides relative to each marked angle.
a) b) c) d)
Practice: Label the sides of the triangles below, then complete the following statements
a) In triangle JKL….
• the length of the hypotenuse is _______________
• the length of the opposite side is ______________
• the length of the adjacent side is ______________
b) In triangle WXY….
• the length of the hypotenuse is _______________
• the length of the opposite side is ______________
• the length of the adjacent side is ______________
SIDE LENGTHS AND ANGLES
a) Look at the triangles below, considering the side lengths and the indicated angle.
Which two triangles are similar? ________________________
| |A |
|Think/Pair/Share: |Think/Pair/Share: |
| | |
|The angles in triangle A are the same in triangle C |Only in similar triangles, are the angles the same |
| | |
|True or false |True or false |
| | |
|Explain: |Explain: |
| | |
| | |
| | |
| | |
RATIO OF SIDE LENGTHS
a) Label the side lengths
b) State the ratio of the opposite side to the adjacent side
| | |A |
TANGENT RATIO
The official name for the ratio between the opposite side and the adjacent side is the TANGENT RATIO
Tangent ratio of the indicated angle = opposite or
adjacent
Because the ratio between the sides length determines the angle, mathematicians have recorded every possible ratio along with its angle and put them in an easy to read chart for you.
|Angle |Tangent |
Therefore we need other names for these ratios.
• The ratio between the OPPOSITE SIDE and the ADJACENT SIDE is called the _________________
• The ratio between the OPPOSITE SIDE and the HYPOTENUSE is called the _____________________
• The ratio between the ADJACENT SIDE and the HYPOTENUSE is called the _____________________
How are we going to remember this??
______________________________________
sin B = cos B = tan B =
Practice: Which primary trig ratio would you choose to help you determine the angle?
(Hint: Label the sides first :D)
a) b) c)
[pic] [pic]
________________ _______________ __________________
The Primary Trig Ratios compare an angle in a right angle triangle to the ratio of two of it’s sides.
How we label our triangle depends on where the given angle is. However, the hypotenuse is always across from the right angle. Label the hypotenuse first!
[pic] [pic] [pic]
1) Write the primary trig ratios for angles A and B of the right angled triangle.
Chart vs Calculator
1. Use your chart to find the measure of each angle to the nearest degree.
(a) sin A = 0.7193 ................
................
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