11-2: The Pythagorean Theorem
Objective:
• You will be able to solve problems using the Pythagorean Theorem.
Add the number of square units within squares A, B, and C.
C
A
B
If we take a right triangle and create squares off each side, the area of the two smaller squares will add up to the area of the larger square. This will happen for ALL right triangles!
Who is Pythagoras?
• a Greek mathematician and philosopher When did he live?
• from 569 to 475 BC
Where did he live?
• born on the island of Samos (a Greek island in the North Agean Sea), moved to Croton, Italy (at the bottom of the “boot”) where he formed a secretive religious brotherhood that furthered mathematical findings
Why is he famous?
• for proving the relationship between the sides of any right triangle • for discovering irrational numbers,
• for proving how to calculate the total number of degrees on the interior of a polygon, • for proving the sum of the outer angles of any polygon is 360 degrees, • for discovering the proportional relationship between the length of a string and its pitch, • for teaching that the Earth and all planets are spheres and their orbits are circular
Vocabulary:
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Angles are labeled with CAPITAL letters.
Sides are labeled with lower case letters.
The sides are labeled with the same letter as the angle that is across the triangle from it.
Investigation:
Refer back to the right triangle activity…
Observations:
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THE PYTHAGOREAN THEOREM:
c
a
b
The hypotenuse is ALWAYS the longest side!
Examples:
Use the Pythagorean Theorem to answer the questions.
1.
What is the length of the hypotenuse of the triangle shown?
c
12 cm
9 cm
Practice Problems:
Use the triangle shown to find the length of the missing side. (Round to the nearest tenth if necessary.)
3.
c
12
35
4.
b
10
26
26 is the LONGEST side... make sure it goes in for "c" in the equation.
5.
40
24
a
40 is the LONGEST side... make sure it goes in for "c" in the equation.
30 ft
b
16 ft
10 ft
Save the extra 10 feet above ground for now...
add it on at the end of the problem.
6.
Suppose you are making a sail in the shape of a right triangle for a sailboat. The length of the longest side of the sail is 65 feet. The sail is to be 63 feet high. What is the length of the third side of the sail?
Always make a right triangle FIRST.
Then label the sides they give you... there will be one side missing. Calculate it's length.
7.
8.
A 20­ft­long wire is used to support a television antenna. The wire is connected to the antenna 15 feet above the ground. How far away from the base of the tower will the other end of the wire be located?
Suppose you leave your house and travel 13 mi due west. Then you travel 3 mi due south. How far are you from your house?