The
PYTHAGOREAN
THEOREM
Red Encabo - Warren Mott
Stephen Lewandowski - Sterling Heights
PURPOSE
1
COMMON CORE STATE STANDARDS
Math.Content.8.G.B.6
Explain the proof of the
Pythagorean Theorem and its
converse.
2
COMMON CORE STATE STANDARDS
Math.Content.8.G.B.7 Apply the
Pythagorean Theorem to determine unknown
side lengths in right triangles in real-world and
mathematical problems.
3
COMMON CORE STATE STANDARDS
Math.Content.8.G.B.8 Apply the
Pythagorean Theorem to find the distance
between two points.
4
REAL LIFE APPLICATIONS
http://images.tutorvista.
com/cms/images/67/pythagorean-word-problem.
png
http://concepts.ck12.org/preview/applicationsusing-the-pythagorean-theorem.jpg
5
TERMS
6
VOCABULARY
RIGHT
TRIANGLE
RIGHT TRIANGLE
triangle with a right, or 90°, angle
triangle with an
angle with a
measure of 90°
2
7
VOCABULARY
HYPOTENUSE
RIGHT TRIANGLE
triangle with a right, or 90°, angle
longest side of
the triangle
opposite of the
right angle
8
2
VOCABULARY
LEGS
OF A TRIANGLE
RIGHT TRIANGLE
triangle with a right, or 90°, angle
the two
shorter sides
of a triangle
legs a and b are
a
shorter than the
hypotenuse
b
9
2
VOCABULARY
ACUTE
TRIANGLE
RIGHT TRIANGLE
triangle with a right, or 90°, angle
triangle with
30°
all angles less
than 90°
75°
75°
10
2
VOCABULARY
OBTUSE
TRIANGLE
RIGHT TRIANGLE
triangle with a right, or 90°, angle
triangle with an
angle greater
than 90°
110°
11
2
PROOF OF THE PYTHAGOREAN THEOREM
2
2
2
a +b =c
the area of a +
the area of b =
the area of c
http://www.regentsprep.
org/Regents/math/geometry/GP13/pythboxes.
gif
12
PYTHAGOREAN TRIPLES
three positive
integers that fit in
the following
equation
2
2
2
a +b =c
http://www.regentsprep.
org/Regents/math/geometry/GP13/pythboxes.
gif
i.e. 3, 4, 5
13
PROBLEM GIVEN TWO LEGS
2
2
2
a +b =c
2
2
2
3 +4 =x
2
3
9 + 16 = x
x2 = 25
x = 5, -5 units
x
4
14
PROBLEM GIVEN ONE LEG & HYPOTENUSE
2
2
2
a +b =c
2
2
2
6 + x = 10
2
36 + x = 100 6
x2 = 64
x = 8, -8 units
10
x
15
RIGHT, ACUTE, OR OBTUSE?
2
2
2
right
2
2
2
acute
2
2
2
obtuse
a +b =c
a +b <c
a +b >c
16
RIGHT, ACUTE, OR OBTUSE?
2
2
2
a +b =c
2
2
2
9 + 12 = 15
81 + 144 = 225 9
225 = 225
right
15
12
17
FINDING PERIMETERS
2
2
2
a +b =c
2
2
2
3 +x =5
2
9 + x = 25
x2 = 16
x = 4, -4 units
3
5
x
18
FINDING PERIMETERS
p = perimeter
p = 2(a) + 2(b)
p = 2(3) + 2(4)
p=6+8
p = 14 units
3
5
4
19
= 1 unit
FINDING DISTANCES
[INSERT VIDEO HERE]
20
CONCLUSION
PROOF
a2 + b 2 = c 2
● Find a side length given two legs
or a leg and hypotenuse
● Find the type of triangle
c
a
b
● Find perimeters
● Find the distance between two
points in a graph
21
MINI QUIZ
22
WHAT IS THE PROOF OF THE PYTHAGOREAN THEOREM?
2
2
2
a +b =c
c
a
b
23
WHAT IS THE LENGTH OF X?
x
5
a
c
12
24
WHAT IS THE LENGTH OF X?
13 units
5
a
c
12
25
WHAT IS THE PERIMETER OF THE RECTANGLE?
Hint: Find the missing side length first!
15
?
9
26
WHAT IS THE PERIMETER OF THE RECTANGLE?
perimeter = 42 units
15
12
9
27
= 1 cm
a
5 cm
b
c
WHAT IS THE
DISTANCE BETWEEN a
AND c?
Find the distance
between [a and b] and
[b and c] in units.
28
QUESTIONS?
29
WORKS CITED
Bogomolny, Alexander. "Pythagorean Theorem." Pythagorean
Theorem and Its Many Proofs. Alexander Bogomolny, 1996. Web. 23 Feb. 2014.
<http://www.cut-the-knot.org/pythagoras/index.shtml>.
Butler, Douglas. "Integer Lists: Pythagorean Triples." Integer Lists: Pythagorean Triples.
Douglas Butler, n.d. Web. 23 Feb. 2014.
<http://www.tsm-resources.com/alists/trip.html>.
"Pythagorean Theorem Algebra Proof." Pythagorean Theorem Proof. MathsIsFun.com, 2013.
Web. 21 Feb. 2014.
<http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html>.
Shmoop Editorial Team. "Common Core Standards : CCSS.Math.Content.8.G.B.6."Shmoop.com.
Shmoop University, Inc., 11 Nov. 2008. Web. 26 Feb. 2014.
<http://www.shmoop.com/common-core-standards/ccss-8-g-6.html>.
30