1. Pythagorean Theorem
We’ve studied the relationship between interior angles of a triangle, and the exterior angles of a triangle. One thing we haven’t analyzed in a triangle is the relationship between the sides

2. Geo-metry: From the root words, “Geo”, meaning earth or land, and “Metry” meaning to measure.
For early human civilizations, measuring the land was one of the most important applications of mathematics. Here’s a classic conundrum that stumped some ancient farmers.
Suppose you own the entire rectangle plot of land below. You want half of your land for live stock, and half of your land for growing corn. You already know that a straight diagonal line connecting opposite vertices, perfectly divides the land into two equal halves. To keep the livestock contained, you need to build a fence. You already know the length and width of the rectangle, so now you need the length of the diagonal line separating the corn from the livestock. How do we do this without having to measure physically the distance?
40 Units
Livestock
30 Units
???? Units
Corn

3. Because the distance in question is really just the hypotenuse of a right triangle, lets look at the land containing the livestock by itself, which we notice is a right triangle.
40 Units
LEG
Livestock
30 Units
??? Units
LEG
Hypotenuse
Lets be sure that we are communicating about this right triangle using the appropriate vocabulary.
Legs:
There are two legs.
They meet to form the right angle.
They are the two shorter sides.
Hypotenuse:
- The longest side
- The side opposite from the right angle.
- Does not touch the right angle.

4. Because the distance in question is really just the hypotenuse of a right triangle, lets look at the land containing the livestock by itself, which we notice is a right triangle.
40 Units
LEG
Livestock
30 Units
??? Units
LEG
Hypotenuse
So how did our ancient farmers figure out this missing length? It has to do with how all three sides of a right triangle are related, and this is what they discovered…

5. Pythagorean Theorem Proof (turn and talk, 5-8 minutes)
What are the side lengths of each leg?
What is the side length of the hypotenuse?
What type of shape is attached to each side of the triangle?
What is the area of each of those shapes?
How do you determine the area of the shapes attached to the sides of the triangle?
Add the areas of the two smaller squares, and compare that sum, to the area of the largest square.

6. So, the Pythagorean theorem looks like this: a2 + b2 = c2
So what does all this tell us, about the relationship between the three side lengths of a right triangle?
We see that “the sum, of the squares of the legs, is equal to the square of the hypotenuse”
To represent the sides of a right triangle we use, a, b, and c. (either leg
Can be a, or b, but the hy-
potenuse is ALWAYS c!)
So, the Pythagorean theorem looks like
this: a2 + b2 = c2
LEG = A
LEG
= B
Hypotenuse = C

7. a2 + b2 = c2
So back to our farmer fence problem. We know the two legs, and want to find the hypotenuse, so using the above formula…
Substitute the known values
Evaluate the exponents.
Combine like terms and isolate the unknown squared variable. (If the unknown side is the hypotenuse, the “c2” is already by itself)
Square root both sides because this is the inverse of squaring a number.
a2 + b2 = c2
= c2
= c2
2500 = c2
√(2500) = √(c2)
50 = c

9. For which triangle are we searching for a missing hypotenuse, and which one has a missing leg? How can we describe what we do differently when solving for a missing leg, rather than solving for the missing hypotenuse?

10. For which triangle are we searching for a missing hypotenuse, and which one has a missing leg? How can we describe what we do differently when solving for a missing leg, rather than solving for the missing hypotenuse?

11. Now you practice trying to find the missing side.
2.)
3.) Find the distance from the Store to the House
1.)
Group A: #1, 3, 4
Group B: #1, 4, 5
Group C: #3-6
6.) How high is the top of the ladder off the ground?
5.)
4.)

12. Square Roots and Non-Perfect Squares
Other examples of perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, …
We know
X2 = 25
X = 25
X = 5
And because 25 has a whole number square root, we call it a PERFECT SQUARE
A Non-Perfect Square is a number which does not have a square root that is a whole number.
For example, 8 is a non-perfect square because
22=4, and 32=9, and because 8 is in-between the perfect squares of 4, and 9, the square root of 8 lies somewhere between 2 and 3.
The square root of a non-perfect square always results in an IRRATIONAL NUMBER.
Irrational numbers: Numbers that cannot be written as a ratio of integers. In decimal form, irrational numbers are non-repeating, and non-terminating (no repeating pattern, and goes on forever)

13. Approximating Irrational Numbers
Pi is an example of an irrational number you are familiar with.
We often use 3.14 for pi, but pi≠ 3.14.
3.14 is only an approximation of pi. Pi continues on as …. We must use approximations of irrational numbers because they never repeat and never end, so we have to stop the number somewhere

14. Approximating the square root of a non-perfect square to the nearest tenth place. Ex: √8 = ?
1.) Which two perfect squares does your non-perfect square lie between? √4 √8 √9 2 ? 3
2.) Find the two square roots of your two perfect squares. Those are the 2 whole numbers your answer is between. So √8 is between 2 and 3, or 2.?
3.) How far is your non-perfect square away from the two perfect squares? This will determine the decimal digit in the tenths place. 8 is one away from 9, and four away from 4. Because 8 is much closer to 9, than 4, we will estimate √8 to be approximately 2.9 or 2.8.

15. Lets try this method… 1.) Find the best approximation of √5

16. Find the missing side x. For any square roots of non-perfect squares, estimate a reasonable decimal answer to the nearest tenth place. A: 1-5, B: 6-10, C: 11-15
#1.)
#2.)
#4.)
#3.)
#5.) 4 24 x x 15 4 x x 8 x 1 12 10 3 6
#6.)
#7.)
#8.)
#9.)
#10.) 24 x 24 25 x 9 x 24 26 4 x 5 12 x 10 10
#11.)
#14.)
#12.)
#13.)
#15.) 36 x 20 x 15 x x 10 x 15 25 12 7 10 x

21. Challenge: More practice finding missing sides
Challenge: More practice finding missing sides. Approximate answers to the nearest tenth place if necessary. (Group A: #1, 2. Group B: #1, 3. Group C: #3, 4)
2.) Find the approximate length of DB
1.) Find the approximate length of AB
3.)
4.) The shape below is a cube. Find the approximate length of the red diagonal line.

22. h = 830 m (height of Burj Khalifa in Dubai,
Center of the Earth
How far is the horizon? Suppose the Earth is an ideal sphere with the radius of 6370 km. How far would you be able to see from a position with a height of:
h = 830 m (height of Burj Khalifa in Dubai,
the tallest human-made structure on Earth)
h = 8848 m (height of Mt. Everest)

23. Proving triangles using the Triangle Inequality Theorem.
This states that the sum of any two sides of a triangle is greater than the length of the third side. (For any triangle)
For example, we would know the three lengths, 3, 4, 8 can not be the sides of any triangle, let alone a right triangle
because < 8 4 3 8
EX1: Can 4, 5, 8 be the three sides of a triangle?
Ex2: Can 3, 7, 12 be the three sides of a triangle?
Ex3: Can 4, 5, 9 be the three sides of a triangle?

24. Converse of Pythagorean Theorem
The Pythagorean Theorem tells us that, “for all right triangles, the sum of the squares of the legs, is always equal to the square of the hypotenuse.
The converse statement is the same idea, but in reverse. Here’s what it says: If, for a triangle, the sum of the squares of the 2 shorter sides is equal to the square of the longest side, than the triangle must be a right triangle (has a right angle).

25. This gives us one of two scenarios…
Either a2 + b2 = c2 and therefore have a right triangle (angle)
OR… a2 + b2 ≠ c2 and therefore do not have a right triangle (angle)

26. Examples:
We know that, both sets of 3 sides lengths will form triangles, (3, 4, 5), and (5, 6, 7) because…
3 + 4 > 5
5 + 6 > 7
Determine now if the above sets of 3 side lengths form a triangle containing a right angle, using the converse of the Pythagorean Theorem…

27. 1.) Prove whether or not the set of 3 given numbers could be the sides lengths of a triangle using the Triangle Inequality Theorem. Justify your answer with work. 2.) For the sets of 3 sides that do make a triangle, prove if they are right triangles, by using the converse of the Pythagorean Theorem. Then answer written response at bottom of slide.
Group A Group B Group C
1.) 1, 2, 6 2.) 6, 8, 10 3.) 8, 15, 17 4.) 1, 1, ) 4, 5, 10 6.) 5, 12, 13 7.) 7, 7, 15 8.) 2, 5, 9 9.) 1.5, 2, 3
1.) 4, 5, 6
2.) 9, 12, 15
3.) 18, 24, 30
4.) 1, 1, 2
5.) 6, 12, 13
6.) 16, 30, 34
7.) 7, 24, 25
8.) 2, 5, 7
9.) 1.5, 2, 2.5
1.) 9, 10, 18
2.) 27, 36, 45
3.) 0.75, 1, 1.25
4.) 1, 2, 3.5
5.) 12, 22.5, 25.5
6.) 7.5, 18, 19.5
7.) 7.5, 17.5, 25
8.) 4.3, 4.5, 8.9
9.) 20, 21, 29
All groups, Written Response: In your own words, how do you know if 2 sides of a triangle form a right angle?

29. Finding the Distance Between Two Points on a Coordinate Plane

31. How do we determine the distance between any two points on the coordinate plane?
On the coordinate plane, why is it easier for us to determine the distance between two points horizontally (AB) or vertically from each other (CD), than it is for us to find the distance between the two points (EF)?

32. The Distance Formula (From the Pythagorean Theorem)
From a2 + b2 = c2 we get…

33. Practice finding the distance between two points.
Group A: Find the distance between point A and B.
Group B: Find the length of line segment CD.
Group C: Determine which side length is longer, AD or AB. D B

34. Put the distances assigned to your group in order from least to greatest. Approximate distances to nearest hundredth.
A: IH, JL, ND,
Explain: Describe all the steps you take to find the distance between two points.
B: AG, DG, and PQ.
Note: P(-40, -53), and
Q(-28, -50).
Explain: When do you think the distance formula becomes better to use than the Pythagorean theorem?
C: PQ, RS, AG
Note: P(-40, -53),
Q(-28, -50), R(30, 3), S(19, -1), Investigate: For right triangles where the sum of the two legs is the same, is the hypotenuse longer when there is a bigger difference between the lengths of the legs or a smaller difference between the lengths of the legs? How could you investigate, and test this out? A C B D I J K L N H Z M G E F

35. Home Work
HW#1: p20-22 read all, do quickchecks #1, 2 HW#2: p22, #3-11 (odd number problems only) must show substitution, and further steps to find hypotenuse. #3: p22, #12-14 #4: p23, #15, 19-21, 28 #5: p27-28, read all, do quickchecks #1, 2. #6: p29, #5, 12, 13 #7: p29-30, #14, 17, 18 #8: p31-32, read all, do quickchecks #1-3 #9: p33, #13-15, 17

36. Quiz on Pythagorean Theorem
Wednesday, 4/5 all classes
4 questions
Students will need to demonstrate…
How to apply the Pythagorean theorem to solve for either a missing leg, or hypotenuse of a right triangle.
How to apply the Triangle Inequality Theorem, and the Converse of the Pythagorean Theorem to prove whether three given side lengths will form a right triangle.
How to determine the distance between two points on a coordinate plane, using either the distance formula, or the Pythagorean theorem.