19.2 Pythagorean Theorem

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19.2 Pythagorean Theorem

A

right

triangle is a triangle that has a right (90 degree) angle. The

2 sides that form the right angle are called the

legs

of the triangle and the side opposite the right angle is called the

hypotenuse

.

hypotenuse leg leg leg leg hypotenuse

The Pythagorean Theorem relates the sides of any right triangle. If

a

and

b

represent the legs and

c

the hypotenuse, then

c

2

a

2

b

2

or

c

a

2

b

2

Find the length of the hypotenuse:

24 dm c

10 dm

Find the length of the hypotenuse:

8.00 cm c

3.90 cm

Find the length of the missing side

25.0 m

18.0 m b

Find the length of the missing side

37,800 ft

42,600 ft b

•How long must a guy wire be to reach from the top of a 15-m telephone pole to a point on the ground 10 m from the foot of the pole?

•Use the Pythagorean theorem.

a

2

b

2

c

2

15

2

10

2

c

2

c

2

325

c

2

325

c

18.028

c

15 m

c

10 m

There are two special right triangles that we will further explore. They are called a 45, 45,

90 right triangle and a 30, 60 ,90 right triangle. The numbers refer to the measure of the angles in degrees.

The 45, 45, 90 triangle is an isosceles triangle which means the two legs have equal measure.

leg leg

45 45 hypotenuse

Since the legs are equal, by the Pyth.Theorem, the hypotenuse has length = leg

2

Find the hypotenuse of an isosceles triangle that has equal sides of 2 m.

2 m

2 m

?

Hypotenuse has length =

2 2

2 .

828

If the hypotenuse of a 45, 45, 90 triangle is 5 cm long, how long is each leg.

?

Since hyp = leg

2

5 cm

5

leg

2

5

leg

2

3 .

536

leg

An equilateral triangle has 3 equal sides and 3 equal angles that each measures 60 degrees. The height bisects both the angle and opposite side making a 30, 60, 90 triangle.

30

60

½ side or ½ hypot

60

Applying the Pythagorean Th. to this 30, 60, 90 triangles gives the following relations:

x

3 other leg

30

2x

, side opposite right angle or longest side

60 x

, side opposite 30 or short leg

Find the missing measures.

Other leg = short leg

3

7 3

12 .

1

in

?

30

Hyp. = 2(short leg) = 2(7 in) = 14 in

?

7 in

60

Find the missing measures.

60

?

30

8 cm

Hyp. = 2(short leg) = 2( 4.6in) = 9.2 in

?

Other leg = short leg

8

3

Shortleg

8

3

4 .

6

shortleg shortleg

3

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