1. The Pythagorean Theorem
Lesson 7-2
The Pythagorean Theorem
Lesson 7-2: The Pythagorean Theorem

2. Anatomy of a right triangle
The hypotenuse of a right triangle is the longest side. It is opposite the right angle.
The other two sides are legs. They form the right angle.
leg
leg
hypotenuse
Lesson 7-2: The Pythagorean Theorem

3. The Pythagorean Theorem
Draw a right triangle with lengths
a, b and c. (c the hypotenuse)
2. Draw a square on each side of
the triangle.
3. What is the area of each square? a b b2 c c2
Lesson 7-2: The Pythagorean Theorem

4. The Pythagorean Theorem
Theorem says
a2 + b2 = c2 a b b2 c c2
Lesson 7-2: The Pythagorean Theorem

5. Proofs of the Pythagorean Theorem
Lesson 7-2: The Pythagorean Theorem

6. The Pythagorean Theorem
If a triangle is a right triangle, with leg lengths a and b and hypotenuse c,
then a2 + b2 = c2
c is the length of the hypotenuse! c a b
Lesson 7-2: The Pythagorean Theorem

7. The Pythagorean Theorem
If a triangle is a right triangle,
then leg2 + leg2 = hyp2
hyp
leg
leg
Lesson 7-2: The Pythagorean Theorem

8. Lesson 7-2: The Pythagorean Theorem
Example
In the following figure if a = 3 and b = 4, Find c.
leg2 + leg2 = hyp2
= C 2
= C2
25 = C2
5 = C c a b
Lesson 7-2: The Pythagorean Theorem

9. Pythagorean Theorem : Examples
c = 17
a = 8, b = 15, Find c
a = 7, b = 24, Find c
a = 9, b = 40, Find c
a = 10, b = 24, Find c
a = 6, b = 8, Find c
c = 25 c a
c = 41
c = 26 b
c = 10
Lesson 7-2: The Pythagorean Theorem

10. Finding the legs of a right triangle:
In the following figure if b = 5 and c = 13, Find a.
leg2 + leg2 = hyp2
a = 132
a = 169
a = 144
a2 = 144
a = 12 c a b
Lesson 7-2: The Pythagorean Theorem

11. Lesson 7-2: The Pythagorean Theorem
More Examples:
1) a=8, c =10 , Find b 2) a=15, c=17 , Find b 3) b =10, c=26 , Find a 4) a=15, b=20, Find c 5) a =12, c=16, Find b 6) b =5, c=10, Find a 7) a =6, b =8, Find c 8) a=11, c=21, Find b
b = 6
b = 8
a = 24
c = 25 c a
a = 8.7
c = 10 b
Lesson 7-2: The Pythagorean Theorem

12. A Little More Triangle Anatomy
The altitude of a triangle is a segment from a vertex of the triangle perpendicular to the opposite side.
altitude
Lesson 7-2: The Pythagorean Theorem

13. Lesson 3-1: Triangle Fundamentals
Altitude - Special Segment of Triangle
a segment from a vertex of a triangle perpendicular to the segment that contains the opposite side.
Definition: B A D F
In a right triangle, two of the altitudes are the legs of the triangle. B A D F I K
In an obtuse triangle, two of the altitudes are outside of the triangle.
Lesson 3-1: Triangle Fundamentals

14. Lesson 7-2: The Pythagorean Theorem
Example:
An altitude is drawn to the side of an equilateral triangle with side lengths 10 inches. What is the length of the altitude?
h = 102
h2 = 75
h =
10 in
10 in h ? 5
10 in in
Lesson 7-2: The Pythagorean Theorem

15. The Pythagorean Theorem – in Review
If a triangle is a right triangle, with side lengths a, b and c (c the hypotenuse,)
then a2 + b2 = c2 c a
What is the converse? b
Lesson 7-2: The Pythagorean Theorem

16. The Converse of the Pythagorean Theorem
If, a2 + b2 = c2,
then the triangle is a right triangle. c a
C is the LONGEST side! b
Lesson 7-2: The Pythagorean Theorem

17. Lesson 7-2: The Pythagorean Theorem
Given the lengths of three sides, how do you know if you have a right triangle?
Given a = 6, b=8, and c=10, describe the triangle.
Compare a2 + b2 and c2:
Since 100 = 100, this is a right triangle. =
a2 + b c2 = c a = = b
Lesson 7-2: The Pythagorean Theorem

18. The Contrapositive of the Pythagorean Theorem
If a2 + b2 c2
then the triangle is NOT a right triangle. c a
What if
a2 + b2 c2 ? b
Lesson 7-2: The Pythagorean Theorem

19. The Contrapositive of the Pythagorean Theorem
If a2 + b2  c2
then either, a2 + b2 > c2 or a2 + b2 < c2 c a
What if
a2 + b2 c2 ? b
Lesson 7-2: The Pythagorean Theorem

20. The Converse of the Pythagorean Theorem
If a2 + b2 > c2 , then
the triangle is acute. c a
The longest side is too short! b
Lesson 7-2: The Pythagorean Theorem

21. The Converse of the Pythagorean Theorem
If a2 + b2 < c2 , then
the triangle is obtuse.
The longest side is too long! a b c
Lesson 7-2: The Pythagorean Theorem

22. Lesson 7-2: The Pythagorean Theorem
Given the lengths of three sides, how do you know if you have a right triangle?
Given a = 4, b = 5, and c =6, describe the triangle.
Compare a2 + b2 and c2:
Since 41 > 36, this is an acute triangle.
>
a2 + b c2
> a
> c
> b
Lesson 7-2: The Pythagorean Theorem

23. Lesson 7-2: The Pythagorean Theorem
Given the lengths of three sides, how do you know if you have a right triangle?
Given a = 4, b = 6, and c = 8, describe the triangle.
Compare a2 + b2 and c2:
Since 52 < 64, this is an obtuse triangle.
<
a2 + b c2
<
< b a
< c
Lesson 7-2: The Pythagorean Theorem

24. Describe the following triangles as acute, right, or obtuse
1) 9, 40, 41 2) 15, 20, 10 3) 2, 5, 6 4) 12,16, 20 5) 14,12,11 6) 2, 4, 3 7) 1, 7, 7 8) 90,150, 120
right
obtuse
obtuse
right c
acute a
obtuse
acute b
right
Lesson 7-2: The Pythagorean Theorem

25. Application
The Distance
Formula

26. The Pythagorean Theorem
For a right triangle with legs of length a and b and hypotenuse of length c, or

27. The x-axis
Start with a horizontal number line which we will call the x-axis.
We know how to measure the distance between two points on a number line. x
Take the absolute value of the difference: │a – b │
│ – 4 – 9 │= │ – 13 │ = 13

28. The y-axis Add a vertical number line which we will call the y-axis.
Note that we can measure the distance between two points on this number line also. y x

29. The Coordinate Plane
We call the x-axis together with the y-axis the coordinate plane. y x

30. Coordinates / Ordered Pair
Coordinates – numbers that identify the position of a point
Ordered Pair – a pair of numbers (x-coordinate, y-coordinate) identifying a point’s position
Identify some coordinates and ordered pairs in the diagram.
Diagram is from the website .

31. Finding Distance in The Coordinate Plane
We can find the distance between any two points in the coordinate plane by using the Ruler Postulate AND the Pythagorean Theorem. y ? x

32. Finding Distance in The Coordinate Plane cont.
First, draw a right triangle. y ? x

33. Finding Distance in The Coordinate Plane cont.
Next, find the lengths of the two legs.
First, the horizontal leg:
│(– 4) – 8│= │– 12│ = 12 y 12 ?
– 4 8 x

34. Finding Distance in The Coordinate Plane cont.
So the horizontal leg is 12 units long.
Now find the length of the vertical leg:
│3 – (– 2)│= │ 5 │ = 5 y 3 ? 5 x
– 2 12

35. Finding Distance in The Coordinate Plane cont.
Here is what we know so far.
Since this is a right triangle, we use the Pythagorean Theorem. y
The distance is 13 units. 13 ? 5 x 12

36. The Distance Formula distance =
Instead of drawing a right triangle and using the Pythagorean Theorem, we can use the following formula:
distance =
where (x1, y1) and (x2, y2) are the ordered pairs corresponding to the two points.
So let’s go back to the example.

37. Example
Find the distance between these two points.
Solution:
First: Find the coordinates of each point. y
(8, 3) 3 ?
– 4 x 8
– 2
(– 4, – 2)

38. Example
Find the distance between these two points.
Solution:
First: Find the coordinates of each point.
(x1, y1) = (-4, -2)
(x2, y2) = (8, 3) y
(8, 3) ? x
(– 4, – 2)

39. Example cont. Solution cont. Then: Since the ordered pairs are
(x1, y1) = (-4, -2) and (x2, y2) = (8, 3)
Plug in x1 = -4, y1 = -2, x2 = 8 and y2 = 3 into
distance = = =