9.2:%20%20The%20Pythagorean%20Theorem

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D.N.A.
1) Find the geometric mean of 8 and 12.
2) The geometric mean of 8 and x is 11. Find x.
Simplify each expression.
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8 2 The Pythagorean Theorem

• Textbook pp. 440 - 446

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Lesson 2 MI/Vocab

• Use the Pythagorean Theorem and its converse.

• Pythagorean triple

Standard 12.0 Students find and use measures of
sides and of interior and exterior angles of
triangles and polygons to classify figures and
solve problems. (Key) Standard 14.0 Students
prove the Pythagorean theorem. (Key) Standard
15.0 Students use the Pythagorean theorem to
determine distance and find missing lengths of
sides of right triangles.
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Pythagorean Theorem

• In a right triangle, the sum of the squares of
  the measures of the legs equals the square of the
  measure of the hypotenuse.

a2 b2 c2
Click here for the Pythagorean Proof
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Lesson 2 CYP2
Find x. Round your answer to the nearest tenth.
A. 17 B. 12.7 C. 11.5 D. 13.2

1. A
2. B
3. C
4. D

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Lesson 2 Ex3
Verify a Triangle is a Right Triangle
COORDINATE GEOMETRY Verify that ?ABC is a right
triangle.
Use distance formula on all 3 sides then the
Pythagorean theorem.
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Lesson 2 Ex3
Verify a Triangle is a Right Triangle
COORDINATE GEOMETRY Verify that ?ABC is a right
triangle.
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Lesson 2 CYP3
COORDINATE GEOMETRY Is ?RST a right triangle?

• A. yes
• B. no
• cannot be determined

1. A
2. B
3. C

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Lesson 2 Ex4
Pythagorean Triples
A. Determine whether 9, 12, and 15 are the sides
of a right triangle. Then state whether they form
a Pythagorean triple.
Since the measure of the longest side is 15, 15
must be c. Let a and b be 9 and 12.
Pythagorean Theorem
Simplify.
Add.
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Homework

• Chapter 8-2
• Pg 444
• 1 3, 6 26

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The Pythagorean Theorem
(Area of green square)
(Area of red square)
Area of the blue square
height

• We start with half the red square, which has
• Area ½ base x height

base

1. We move one vertex while maintaining the base
   height, so that the area remains the same. This
   is called a SHEAR.

base

1. We rotate this triangle, which does not change
   its area.

height

1. We mark the base and height for this triangle.

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The Pythagorean Theorem
(Area of green square)
(Area of red square)
Area of the blue square
height

• We start with half the red square, which has
• Area ½ base x height

Half the red square.
base

1. We move one vertex while maintaining the base
   height, so that the area remains the same. This
   is called a SHEAR.

1. We rotate this triangle, which does not change
   its area.

1. We mark the base and height for this triangle.

1. We now do a shear on this triangle, keeping the
   same area.

Remember that this pink triangle is half the red
square.
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The Pythagorean Theorem
(Area of green square)
(Area of red square)
Area of the blue square

1. The other half of the red square has the same
   area as this pink triangle, so if we copy and
   rotate it, we get this.

So, together these two pink triangles have the
same area as the red square.
Half the red square.
Shear

1. We now take half of the green square and
   transform it the same way.

Half the red square.
Half the green square.
Half the green square.
We end up with this triangle, which is half of
the green square.
Rotate

1. The other half of the green square would give us
   this.

Shear

• Together, they have they same
• area as the green square.

So, we have shown that the red green
squares together have the same area as the blue
square.
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The Pythagorean Theorem
Weve Proven the Pythagorean Theorem (click to
return)
(Area of green square)
(Area of red square)
Area of the blue square

1. The other half of the red square has the same
   area as this pink triangle, so if we copy and
   rotate it, we get this.

So, together these two pink triangles have the
same area as the red square.
Half the red square.

1. We now take half of the green square and
   transform it the same way.

Shear
Half the red square.
Half the green square.
We end up with this triangle, which is half of
the green square.
Half the green square.

1. The other half of the green square would give us
   this.

Rotate

• Together, they have they same
• area as the green square.

Shear
So, we have shown that the red green
squares together have the same area as the blue
square.
Weve PROVEN the Pythagorean Theorem!
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Find the missing side of the triangle.
1)
2)
3)
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Lesson 1 Ex4
Hypotenuse and Segment of Hypotenuse
Find c and d in ?JKL.
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Lesson 1 Ex4
Hypotenuse and Segment of Hypotenuse
Find c and d in ?JKL.
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11.2