Download 5-85 Pythagorean Triples

5.85 Pythagorean Triples Investigation
Name: _____________________________________
Three positive integers that work in the Pythagorean formula, a 2  b 2  c 2 , are called Pythagorean
triples.
To use Pythagoras's Theorem, you must have a ____________________________________ triangle
and need to find the length of a side. Just because three numbers can be used in the equation,
a 2  b 2  c 2 , does that mean those three numbers form a right triangle?
To begin, you will need some string, a ruler, and a protractor. Your goal is to follow the steps and fill
in the chart below.
Step 1:
On a piece of string, mark off the distances of the given Pythagorean triple and make a
triangle.
Using a protractor, measure the largest angle. Did the triple make a right triangle?
Step 2:
Pythagorean Triple
3-4-5
5-12-13
8-15-17
5 cm
3 cm
Did you make a right triangle?
13 cm
5 cm
4 cm
12 cm
Conjecture 1: If the lengths of the three sides of a triangle work in the Pythagorean formula, then the
8 cm
17 cm
5 cm
13 cm
5 cm
triangle
________________________________________.
3 cm
5 cm
3 cm
15 cm
4 cm
13 cm
5 cm
4 cm
12 cm
12 cm
Use the conjecture just made to determine whether
each triangle is a right triangle.
1.
2.
3.
8 cm
15 cm
5 cm
3 cm
5 cm
13 cm
17
3 cm
4 cm
17 cm
5 cm
8
13
13 cm
5 cm
5
4 cm
12 cm
8 cm
12 cm
36
17 cm
3 cm
5 cm
13 cm
5 cm
12
4 cm
15
8 cm
8 cm
12
12 cm
35
17 cm
15 cm
17 cm
8 cm
17 cm
15 cm
15 cm
4.
15 cm
5.
18
12
22
7 cm
20
1.73
2.23
24
15 cm
6.
10
1.41
Use the following triangles for the next investigation.
5 cm
3 cm
13 cm
5 cm
4 cm
12 cm
8 cm
17 cm
15 cm
Step 1:
For each triangle above, add on 2 to the length of each side. Write your new triples
below.
Step 2:
Substitute your new triples into the Pythagorean formula. Which one works?
Step 3:
For each triangle above, double the length of each side. Write your new triples below.
Step 4:
Substitute your new triples into the Pythagorean formula. Which one works?
Step 5:
For each triangle above, triple the length of each side. Write your new triples below.
Step 6:
Substitute your new triples into the Pythagorean formula. Which one works?
Step 7:
For each triangle above, multiply each side by a number of your choice. Write your
new triples below.
Step 8:
Substitute your new triples into the Pythagorean formula. Which one works?
Conjecture 2: Adding the same number to each of the numbers of a Pythagorean Triple
__________________________ create a new Pythagorean Triple.
Conjecture 3: If you multiply the lengths of all three sides of any right triangle by the same number,
the resulting triangle will be a ____________________________________________.
The following are four primitive Pythagorean triples you should either memorize or have on your half
sheet.
3, 4, 5
8, 15, 17
5, 12, 13
7, 24, 25
Each of the following problems involves one of the four most common Pythagorean primitives.
Recognize then and you can save yourself a lot of time and work.
1. a = ?
2. b = ?
3. What is the perimeter?
4. c = ?
5. The area of the rectangle is
168 sq ft. d = ?
6. What is the area of the
rectangle?
7. What is the area of the
square?
8. What is the exact area of the
semicircle?
9. m = ?
10. Complete the table below by creating multiples of the most common Pythagorean primitives.
Primitive
Doubles
Triples
4 times
10 times
3, 4, 5
6, 8, 10
15, 36, 39
32, 60, 68
70, 240, 250
In problems 11 to 20, each right triangle has sides whose lengths are multiples of a Pythagorean
primitive. Check your completed table of multiples of Pythagorean triples to solve for the indicated
values.
21. Find the area of a triangle with side lengths of 20, 20, and 12.
22. Find the value of x.
a)
b)
c)
23. Find the coordinates of point T.
24. Find the coordinates of point S.
25. Solve:
a) x 2  x  30
b) 2 x 2  3x  5
26. Simplify the radicals.
a)
12  2 27
b)
48
c)
2 5
2
d) 2 3  5 8
27. Calculate the value of x.
3x
2x-1
x+6