Unit 3: GeometryLesson #6: Triangles & Pythagorean Theorem

LEARNING GOALS• Find missing sides in right triangles• To determine if a triangle is right or not• To explain Pythagorean theorem• To explain how to and when to use the Pythagorean theorem

3

Classifying Triangles by Sides

Equilateral:

Scalene: A triangle in which all 3 sides are different lengths.

Isosceles: A triangle in which at least 2 sides are equal.

A triangle in which all 3 sides are equal.

AB =

3.0

2 cm

AC = 3.15 cm

BC = 3.55 cm

A

B CAB = 3.

47 cm

AC = 3.47 cm

BC = 5.16 cmB C

A

HI = 3.70 cm

G

H I

GH = 3.70 cm

GI = 3.70 cm

4

Classifying Triangles by Angles

Acute:

Obtuse:

A triangle in which all 3 angles are less than 90˚.

A triangle in which one and only one angle is greater than 90˚& less than 180˚

108 °

44 °

28 °B

C

A

57 ° 47 °

76 °G

H I

5

Classifying Triangles by Angles

Right:

Equiangular:

A triangle in which one and only one angle is 90˚

A triangle in which all 3 angles are the same measure.

34°56°

90°B C

A

60°

60°60°C

B

A

6

polygons

Classification by Sides with Flow Charts & Venn Diagrams

triangles

Scalene

Equilateral

Isosceles

Triangle

Polygon

scalene

isosceles

equilateral

7

polygons

Classification by Angles with Flow Charts & Venn Diagrams

triangles

Right

Equiangular

Acute

Triangle

Polygon

right

acute

equiangularObtuse

obtuse

Pythagoras• Lived in southern Italy

during the sixth century B.C.

• Considered the first true mathematician

• Used mathematics as a means to understand the natural world

• First to teach that the earth was a sphere that revolves around the sun

Right Triangles• Longest side is the

hypotenuse, side c (opposite the 90o angle)

• The other two sides are the legs, sides a and b

• Pythagoras developed a formula for finding the length of the sides of any right triangle

The Pythagorean Theorem

“For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.”

a2 + b2 = c2

It only works with right-angled triangles.

hypotenuse

The longest side, which is always opposite the right-angle, has a special name:

This is the name of Pythagoras’ most famous discovery.

Pythagoras’ Theorem

c

b

a

c²=a²+b²

Pythagoras’ Theorem

c

a

c

c

b

b

b a

a

c

ya

Pythagoras’ Theorem

c²=a²+b²

1m

8m

Using Pythagoras’ Theorem

What is the length of the slope?

1m

8m

c

b=a=

c²=a²+ b²

c²=1²+ 8²c²=1 + 64c²=65

?

Using Pythagoras’ Theorem

How do we find c?

We need to use the square root button on the calculator.It looks like this √Press

c²=65

√ , Enter 65 =

So c= √65 = 8.1 m (1 d.p.)

Using Pythagoras’ Theorem

Example 1

c

12cm

9cm

a

b c²=a²+ b²c²=12²+ 9²

c²=144 + 81

c²= 225 c = √225= 15cm

c

6m4m

s

a bc²=a²+ b²s²=4²+ 6²

s²=16 + 36

s²= 52 s = √52

=7.2m (1 d.p.)

Example 2

7m

5m

hc

a

b

c²=a²+ b²7²=a²+ 5²

49=a² + 25 ?

Finding the shorter side

49 = a² + 25

We need to get a² on its own.Remember, change side, change sign!

Finding the shorter side

+ 25

49 - 25 = a²a²= 24 a = √24 = 4.9 m (1 d.p.)

169 = w² + 36

c

w

6m13m

a

b

c²= a²+ b²13²= a²+ 6²

169 – 36 = a²

a = √133 = 11.5m (1 d.p.)

a²= 133

Example 1

169 = a² + 36

Change side, change sign!

Applications

• The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications.

• The theorem is invaluable when computing distances between two points, such as in navigation and land surveying.

• Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand.

Baseball ProblemA baseball “diamond” is really a square.

You can use the Pythagorean theorem to find distances around a baseball diamond.

Baseball ProblemThe distance between consecutive bases is 90feet. How far does a catcher have to throwthe ball from home plate to second base?

Baseball ProblemTo use the Pythagorean

theorem to solve for x, find the right angle.

Which side is the hypotenuse?

Which sides are the legs?Now use: a2 + b2 = c2

Baseball ProblemSolution• The hypotenuse is the

distance from home to second, or side x in the picture.

• The legs are from home to first and from first to second.

• Solution: x2 = 902 + 902 = 16,200 x = 127.28 ft

Ladder Problem A ladder leans against a

second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house,

how high is the window?

Ladder ProblemSolution

• First draw a diagram that shows the sides of the right triangle.

• Label the sides: • Ladder is 25 m• Distance from house is 7

m• Use a2 + b2 = c2 to solve

for the missing side. Distance from house: 7 meters

Ladder ProblemSolution

72 + b2 = 252

49 + b2 = 625 b2 = 576 b = 24 m

How did you do?A = 7 m

Success Criteria• I can identify a right-angle triangle• I can identify Pythagorean theorem• I can identify when to use Pythagorean theorem• I can use Pythagorean theorem to find the longest side• I can use Pythagorean theorem to find the shortest side• I can solve problems using Pythagorean theorem