1. Example 1
What is the area of a square with a side length of 4 inches? x inches?
What is the side length of a square with an area of 25 in2? x in2? x

2. Parts of a Right Triangle
Which segment is the longest in any right triangle?

3. Apply the Pythagorean Theorem
Objectives:
To discover and use the Pythagorean Theorem
To use Pythagorean Triples to find quickly find a missing side length in a right triangle

4. The Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

5. The Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

6. The Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

7. Example 2
The triangle below is definitely not a right triangle. Does the Pythagorean Theorem work on it?

8. Example 3
How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall?

9. Example 4: SAT
In figure shown, what is the length of RS?

10. Example 5
What is the area of the large square?

11. Example 6
Find the area of the triangle.

12. Pythagorean Triples
Three whole numbers that work in the Pythagorean formulas are called Pythagorean Triples.

13. Example 7
What happens if you add the same length to each side of a right triangle? Do you still get another right triangle?

14. Example 8
What happens if you multiply all the side lengths of a right triangle by the same number? Do you get another right triangle?

15. Pythagorean Multiples
Pythagorean Multiples Conjecture:
If you multiply the lengths of all three sides of any right triangle by the same number, then the resulting triangle is a right triangle.
In other words, if a2 + b2 = c2, then (an)2 + (bn)2 = (cn)2.

16. Pythagorean Triples

17. Primitive Pythagorean Triples
A set of Pythagorean triples is considered a primitive Pythagorean triple if the numbers are relatively prime; that is, if they have no common factors other than 1.
3-4-5

18. Example 9
Find the length of one leg of a right triangle with a hypotenuse of 35 cm and a leg of 28 cm.

19. Example 10
Use Pythagorean Triples to find each missing side length.

20. Example 11
A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the top of the ladder slips down 4 feet, how many feet will the bottom slip out?

21. Converse of the Pythagorean Theorem
Objectives:
To investigate and use the Converse of the Pythagorean Theorem
To classify triangles when the Pythagorean formula is not satisfied

22. 7.2 Converse of the Pythagorean Theorem
Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle.

23. Example 3
Which of the following is a right triangle?

24. Example 4
Tell whether a triangle with the given side lengths is a right triangle.
5, 6, 7
5, 6,
5, 6, 8

25. These numbers are called the Perfect Squares.
Example 1
Write the first 20 terms of the following sequence:
1, 4, 9, 16, … x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x2 1 4 9 16 25 36 49 64 81
100
121
144
169
196
225
256
289
324
361
400
These numbers are called the Perfect Squares.

26. Square Roots The number r is a square root of x if r2 = x.
This is usually written
Any positive number has two real square roots, one positive and one negative, √x and -√x
√4 = 2 and -2, since 22 = 4 and (-2)2 = 4
The positive square root is considered the principal square root

27. Example 2
Use a calculator to evaluate the following:

28. Example 3
Use a calculator to evaluate the following:

29. Properties of Square Roots
Properties of Square Roots (a, b > 0)
Product Property
Quotient Property

30. 7.4 Special Right Triangles
Simplifying Radicals
Objectives:
To simplify square roots
To solve quadratic equations

31. Simplifying Square Root
The properties of square roots allow us to simplify radical expressions.
A radical expression is in simplest form when:
The radicand has no perfect-square factor other than 1
There’s no radical in the denominator

32. Simplest Radical Form
Like the number 3/6, is not in its simplest form. Also, the process of simplification for both numbers involves factors.
Method 1: Factoring out a perfect square.

33. Simplest Radical Form
In the second method, pairs of factors come out of the radical as single factors, but single factors stay within the radical.
Method 2: Making a factor tree.

34. Simplest Radical Form
This method works because pairs of factors are really perfect squares. So 5·5 is 52, the square root of which is 5.
Method 2: Making a factor tree.

35. Investigation 1
Express each square root in its simplest form by factoring out a perfect square or by using a factor tree.

36. Exercise 4a
Simplify the expression.

37. Exercise 4b
Simplify the expression.

38. Example 5
Simplify the expression.

39. Example 6 Evaluate, and then classify the product. (√5)(√5) =
(2 + √5)(2 – √5) =

40. Conjugates are Magic!
The radical expressions a + √b and a – √b are called conjugates.
The product of two conjugates is always a rational number

41. Example 7
Identify the conjugate of each of the following radical expressions: √7
5 – √11
√13 + 9

42. Example 8
Recall that a radical expression is not in simplest form if it has a radical in the denominator. How could we use conjugates to get rid of any damnable denominator-bound radicals?

43. Rationalizing the Denominator
We can use conjugates to get rid of radicals in the denominator: The process of multiplying the top and bottom of a radical expression by the conjugate of the denominator is called rationalizing the denominator.
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44. Exercise 9a
Simplify the expression.

45. Exercise 9b
Simplify the expression.

46. Solving Quadratics
If a quadratic equation has no linear term, you can use square roots to solve it.
By definition, if x2 = c, then x = √c and x = −√c, usually written x = ±√c
You would only solve a quadratic by finding a square root if it is of the form
ax2 = c
In this lesson, c > 0, but that does not have to be true.

47. Solving Quadratics
If a quadratic equation has no linear term, you can use square roots to solve it.
By definition, if x2 = c, then x = √c and x = -√c, usually written x = √c
To solve a quadratic equation using square roots:
Isolate the squared term
Take the square root of both sides

48. Exercise 10a
Solve 2x2 – 15 = 35 for x.

49. Exercise 10b
Solve for x.

50. The Quadratic Formula
Let a, b, and c be real numbers, with a ≠ 0. The solutions to the quadratic equation ax2 + bx + c = 0 are
Song 1:
Song 2:

51. Exercise 11a
Solve using the quadratic formula. x2 – 5x = 7

52. Exercise 11b Solve using the quadratic formula. x2 = 6x – 4
4x2 – 10x = 2x – 9
7x – 5x2 – 4 = 2x + 3

53. Exercise 12 Based on the previous Exercise,
How can the quadratic formula tell you how many solutions to expect?
How can the quadratic formula tell you what kind of solutions to expect: Real or imaginary, rational or irrational?
How are the roots related to each other if they are irrational or imaginary?

54. The Discriminant
In the quadratic formula, the expression b2 – 4ac is called the discriminant.
Discriminant