Lesson 7-2 Lesson 7-2: The Pythagorean Theorem1 The Pythagorean Theorem.

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

Transcript of Lesson 7-2 Lesson 7-2: The Pythagorean Theorem1 The Pythagorean Theorem.

Lesson 7-2

Lesson 7-2: The Pythagorean Theorem 1

The Pythagorean Theorem

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.

Lesson 7-2: The Pythagorean Theorem 2

hypotenuse

leg

leg

The Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem

3

a

bc

1. 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?

a2

b2

c2

The Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem

4

a

bc

The Pythagorean Theorem says

a2 + b2 = c2

a2

b2

c2

The Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 6

a

b

c

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!

The Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 7

leg

leg

hyp

If a triangle is a right triangle,

then leg2 + leg2 = hyp2

In the following figure if a = 3 and b = 4, Find c.

leg2 + leg2 = hyp2

32 + 42 = C 2

9 + 16 = C2

25 = C2

5 = C

Example

Lesson 7-2: The Pythagorean Theorem 8

a

b

c

225 C

Pythagorean Theorem : Examples

1. a = 8, b = 15, Find c

2. a = 7, b = 24, Find c

3. a = 9, b = 40, Find c

4. a = 10, b = 24, Find c

5. a = 6, b = 8, Find c

Lesson 7-2: The Pythagorean Theorem 9

a

b

c

c = 17

c = 25

c = 41

c = 26

c = 10

In the following figure if b = 5 and c = 13, Find a.

leg2 + leg2 = hyp2

a2 +52 = 132

a2 + 25 = 169

-25 -25

a2 = 144

a2 = 144

a = 12

Finding the legs of a right triangle:

Lesson 7-2: The Pythagorean Theorem 10

a

b

c

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

Lesson 7-2: The Pythagorean Theorem 11

a

b

c

b = 6

b = 8

a = 24

c = 25

a = 8.7

c = 10

112b

320 8 5b

A Little More Triangle Anatomy

• The altitude of a triangle is a segment from a vertex of the triangle perpendicular to the opposite side.

Lesson 7-2: The Pythagorean Theorem 12

altitude

B

A DE

C

F, , .AF BE DC are the altitudes of the triangle

, ,AB AD AF altitudes of right

, ,BI DK AF altitudes of obtuse Lesson 3-1: Triangle Fundamentals 13

Altitude - Special Segment of Triangle

Definition: a segment from a vertex of a triangle perpendicular to the segment that contains the opposite side.

In a right triangle, two of the altitudes are the legs of the triangle.

B

A D

F

In an obtuse triangle, two of the altitudes are outside of the triangle.

B

A D

F

I

K

Example:

• An altitude is drawn to the side of an equilateral triangle with side lengths 10 inches. What is the length of the altitude?

Lesson 7-2: The Pythagorean Theorem 14

h2 + 52 = 102

h2 = 75

h =

10 in 10 in

10 in

h

?5

75

25 3 5 3 in

The Pythagorean Theorem – in Review

Lesson 7-2: The Pythagorean Theorem 15

a

b

c

Pythagorean Theorem:

If a triangle is a right triangle, with side lengths a, b and c (c the hypotenuse,)

then a2 + b2 = c2

What is the converse?

The Converse of the Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 16

a

b

c

If, a2 + b2 = c2,

then the triangle is a right triangle.

C is the LONGEST side!

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 + b2 c2

62 + 82 102

36+ 64 100100 100

=

Given the lengths of three sides, how do you know if you have a

right triangle?

Lesson 7-2: The Pythagorean Theorem 17

a

b

c ===

The Contrapositive of the Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 18

a

b

c

If a2 + b2 c2

then the triangle is NOT a right triangle.

What if a2 + b2 c2

?

The Contrapositive of the Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 19

a

b

c

If a2 + b2 c2

then either, a2 + b2 > c2 or a2 + b2 < c2

What if a2 + b2 c2

?

The Converse of the Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 20

a

b

c

If a2 + b2 > c2 , then

the triangle is acute.

The longest side is too short!

The Converse of the Pythagorean Theorem

Lesson 7-2: The Pythagorean Theorem 21

a b

c

If a2 + b2 < c2 , then

the triangle is obtuse.

The longest side is too long!

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 + b2 c2

42 + 52 62

16 + 25 3641 36

>

Given the lengths of three sides, how do you know if you have a

right triangle?

Lesson 7-2: The Pythagorean Theorem 22

a

b

c>>>

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 + b2 c2

42 + 62 82

16 + 36 6452 64

<

Given the lengths of three sides, how do you know if you have a

right triangle?

Lesson 7-2: The Pythagorean Theorem 23

a b

c

<<<

Describe the following triangles as acute, right, or obtuse

1) 9, 40, 412) 15, 20, 103) 2, 5, 64) 12,16, 205) 14,12,116) 2, 4, 37) 1, 7, 78) 90,150, 120

Lesson 7-2: The Pythagorean Theorem 24

a

b

c

right

acute

obtuse

obtuse

right

right

obtuse

acute

ApplicationThe Distance

Formula

The Pythagorean Theorem

• For a right triangle with legs of length a and b and hypotenuse of length c,

2 2c a b

2 2 2c a b or

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

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.

x

y

The Coordinate Plane

x

y

We call the x-axis together with the y-axis the coordinate plane.

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 www.ezgeometry.com .

Finding Distance inThe Coordinate Plane

x

y

We can find the distance between any two points in the coordinate plane by using the Ruler Postulate AND the Pythagorean Theorem.

?

Finding Distance inThe Coordinate Plane cont.

x

yFirst, draw a right triangle.

?

Finding Distance inThe Coordinate Plane cont.

x

y

Next, find the lengths of the two legs.

•First, the horizontal leg:

?– 4 8

│(– 4) – 8│= │– 12│ = 12

12

Finding Distance inThe Coordinate Plane cont.

x

y

So the horizontal leg is 12 units long.

•Now find the length of the vertical leg:

?3

– 2

│3 – (– 2)│= │ 5 │ = 5

12

5

Finding Distance inThe Coordinate Plane cont.

x

y

?

12

5

Here is what we know so far.

Since this is a right triangle, we use the Pythagorean Theorem.

2 2c 5 12 169

13 13

The distance is 13 units.

The Distance Formula

2 2

2 1 2 1x x y y

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.

Example

x

y

Find the distance between these two points.

Solution:

First: Find the coordinates of each point.

?

– 2

– 4

(– 4, – 2)

8

3

(8, 3)

Example

x

y

Find the distance between these two points.

Solution:

First: Find the coordinates of each point.

(x1, y1) = (-4, -2)

(x2, y2) = (8, 3)

?

(– 4, – 2)

(8, 3)

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 =

=

=

= 13

Example cont.

2 2

2 1 2 1x x y y

2 28 4 3 2

2 212 5