8 Grade
Common Core
Math
th
Booklet 3
Geometry
One of the Main Ideas in Geometry:
Understand and apply the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean Theorem states that on a right triangle, the length of the
short leg squared plus the length of the long leg squared equals the
hypotenuse (longest side) squared.
It is written like this: a2 + b2 = c2
Where a is the shortest leg, b is
the longer leg, and c is the hypotenuse.
c
a
b
Let’s do an example of the Pythagorean Theorem.
If leg a is 3 and leg b is 4, what would side c be?
If we plug in 3 for a and 4 for b using the formula for the Pythagorean Theorem,
we see that 32 + 42 = c2
That means 9 + 16 = c2 so c2 = 25
If we find the square root of c2, then we see that c = √25
So c = 5
REMEMBER
*This Theorem only works on RIGHT TRIANGLES*
8th Grade Common Core Math Standards:
Standard 8.G.B.6: Explain a proof of the Pythagorean Theorem and its
converse. (Students must prove why the Pythagorean Theorem works to
find the length of the hypotenuse of a right triangle and why it works to
also find the length of either leg of a right triangle.)
Standard example:
Deriving the equation of the Pythagorean Theorem a2 + b2 = c2
In order to prove that a2 + b2 = c2 we will use this
square figure on the left.
The area of the total figure is (a+b) * (a+b) which is
the length of the larger square times the width of
the larger square.
The area of the smaller square is c * c which is c2
!
The area of one triangle is (a*b). Since there are
!
!
4 triangles we multiply !(a*b) by 4 to get a total
area of 2ab.
Distributive Property Tutorial
(a + b) * (a + b)
Order in Which you multiply
1.
2.
3.
4.
Red [a*a] ! a2
Green [a*b] ! ab
Blue [b*a] ! ab
Black [b*b] ! b2
So the full equation becomes
a2 + 2ab + b2
The area of the whole figure then is the area of the
smaller square plus the total area of the four
triangles: c2 (smaller square) + 2ab (triangles) =
(a+b) * (a+b) which represents the area of the
whole figure.
c2 + 2ab = (a+b) * (a+b)
Now let’s multiply (a+b) * (a+b) by using the
distributive property of multiplication (a+b) * (a+b)
= a*a + a*b + b*a + b*b
or a2 + 2ab + b2
Now we know a2 + 2ab + b2 = c2 + 2ab
If we subtract 2ab from both sides, we end up with
the Pythagorean Theorem
a2 + b2 = c2
Standard 8.G.B.7: Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in real-world and mathematical
problems in two and three dimensions.
Standard example:
Question: Find the missing length using the Pythagorean
Theorem
Pythagorean Theorem: a2 + b2 = c2
a=8
?
b = 12
Answer: 82 + 122 = c2
64 + 144 = c2
12
208 = c2
c = √208
8
Since √208 is not a perfect square root, we simplify by
finding two numbers that when multiplied together equal
208.
One of the numbers must be a perfect square.
We see that 16 and 13 when multiplied together equal
208.
16 is a perfect square. So √16 x √13 = √208 .
If we simplify further, the √16 = 4 so 4 x √13 = √208
c = 4√13
Standard examples continued:
b = 12
13
?
12
c = 13
Answer: To find the value of side “a” we
plug the values of side “b” and “c” into the
formula a2 + b2 = c2
a2 + 122 = 132
(subtract 122 from both sides of the
equation)
a2 = 132 - 122
a2 = 169 - 144
a2 = 25
a = √25
a=5
Standard 8.G.B.8: Apply the Pythagorean Theorem to find the distance
between two points in a coordinate system (two points on a graph).
Standard example: Find the distance between two points.
12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Question: What is the distance between point (1,4) and point (4,10)?
Answer: First, We draw a right triangle between the two points
12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 We see that the black line is 3 units long and the red line is 6 units tall. To find the
distance between the two points we use the Pythagorean Theorem: a2 + b2 = c2
32 [black side] + 62 [red side] = Distance2 [blue side]
9 + 36 = Distance2 units
45 = Distance2 units
Distance = √45 units
Since √45 is not a perfect square root, we simplify by finding two numbers that
when multiplied together equal 45 where one of the numbers is a perfect square.
9 * 5 = 45 and 9 is a perfect square.
So, the √45 can be simplified to √9 * √5
Distance = 3√5 units
WHY THIS IS IMPORTANT
The Pythagorean Theorum is important because it is used all the time in
every day life. For example, a painter painting a two-story house would
use the Pythagorean Theorum or its converse to determine the size of the
ladder he needs to use, or the distance he needs to place the base of the
ladder away from the house to safely be able to climb it without it tipping
over (and reaching the highest point of the house).
You may also may use the Pythagorean Theorum when purchasing a flat
screen TV that needs to fit inside a rectangular cabinet. For example, if
your cabinet is 32” x 24” wide, you will need to find the hypotenuse of your
cabinet when purchasing your TV since the size of the TV is measured
from corner to corner (hypotenuse) to make sure the TV fits inside your
cabinet.