The Pythagorean Theorem & Special Right Triangles...The Pythagorean Theorem & Special Right Triangles We are all familiar with the Pythagorean Theorem and now we’ve explored one

Chapter 7: Right Triangles & Trigonometry Name _____________________________ Sections 1 – 4 Geometry Notes

The Pythagorean Theorem & Special Right Triangles We are all familiar with the Pythagorean Theorem and now we’ve explored one proof – there

are 370 known proofs, by the way! – let’s put it in to practice.

The

orem

7.1

Pythagorean Theorem

In a __________________ triangle, the _______________ of

the length of the ______________________ is equal to

the _________ of the __________________ of the lengths

of the ____________.

Refresh your memory: the hypotenuse is _______________________________________

And, the legs are _______________________________________________________________

Radical, dude.

Since we will be dealing with square __________, we want to also refresh our skills in this area.

¿ First: does every number have a square root?

____________, but remember! It may not be a ______________ number.

Estimates versus simplified radicals

You should be pretty good at problems like these:

36 = ____________ OR 49 = ____________ OR 121 = ____________

How do you solve something like this without a calculator? 28

• Let’s break it down:

This answer, _________________, is actually the ________________answer, whereas what you get

from a calculator, ____________________, is only an _______________________.

Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4

Page 2 of 12

Try some more:

98 50 216

Estimate: ________________ Estimate: ________________ Estimate: ________________

A little more radical practice …

23 27x 24 256x 27 70x

How about this one? 53

… you can’t leave a radical in the denominator…


Page 3 of 12

Try this:

AB = 8, BC = 6, AC = ?

Can you think of 3 other side lengths that will come out to be “perfect” like the one above?

(Hint: look at the side lengths above and see if they are multiples or factors of similar numbers.)

These special sets of positive ____________________ are called _________________________

Triples and can be used to make ___________________ angles where there are none.

Practice 7.1

Page 436 – 7: 3 – 5, and more ….


Page 4 of 12

Page 436 – 7: 8 – 10, 14 – 17


Page 5 of 12

The Converse of the Pythagorean Theorem

Up to now, we’ve said the Pythagorean Theorem, ______ + _______ = ________, is used only with

right triangles. As you might have suspected, a version of it can also be used with all other

triangles.

Try it!

Complete the chart below by finding values for c that make the equation/inequality true.

The catch! c must be greater than either a or b, but less than a + b.

Equation / Inequality a = b = a + b √ c = ? (> a, > b, < a + b)

a2 + b2 = c2 6 8

a2 + b2 < c2 6 8

a2 + b2 > c2 6 8

a2 + b2 = c2 5 12

a2 + b2 < c2 5 12

a2 + b2 > c2 5 12

a2 + b2 = c2 9 12

a2 + b2 < c2 9 12

a2 + b2 > c2 9 12

2. Construct these triangles; you may use Patty Paper or simply draw them on scrap / white paper.

3. Make a conjecture about the type of triangle that results for each of the following possibilities:

a. a2 + b2 = c2 _______________________________________________________

b. a2 + b2 < c2 _______________________________________________________

c. a2 + b2 > c2 _______________________________________________________


Page 6 of 12

The

orem

7.2

Converse of the Pythagorean Theorem

If the sum of the _____________ of the lengths of two sides of a triangle

_______________ the square of the length of the third side, then the triangle is a

___________________ triangle and the longest side is the _______________________.

The

orem

s 7.

3 &

7.4

Pythagorean Inequality Theorems

If the sum of the __________________ of the lengths of the _______________ two sides

of a triangle is ____________________ than the square of the length of the longest

side, then the triangle is _________________.

If the sum of the squares of the lengths of the shorter ___________ sides of a

triangle is __________ than the square of the length of the _______________________

side, then the triangle is ________________________.

Practice 7.2

Page 444: 3 – 5, and more ….


Page 7 of 12

Page 444: 9 – 23 ODDS


Page 8 of 12

Special Right Triangles

What do you get when you cut a square in half?

An _________________ _________________

triangle, also called a

_______ – _______ – _______

because of its angle measurements.

Why so special? Complete the chart.

Let’s do some calculating: find the length of the hypotenuse of the isosceles right triangle

using the given values. Keep your answer in the simplest radical form.

Leg (a) Leg (b) Hypotenuse (c)

3

4

5

6

12

x

Work area

Notice anything?

a

b

c


Page 9 of 12

What do you get when you cut an equilateral triangle in half?

An _________________ _________________

triangle, also called a

_______ – _______ – _______

because of its angle measurements.

Why so special?

Let’s start with some deductive thinking. Triangle ABC is equilateral; CD is an altitude.

1. What are m Aand m B ?

2. What are m ACD and m BCD ?

3. What are m ADC and m BDC?

4. Is ACD BCD ? Why?

5. Is AD BD ? Why?

Note that altitude CD divides the equilateral triangle into two right triangles with acute

angles of _________ and _________.

Look at just one of the _______ – _______ – _______ triangles and compare the leg and the

hypotenuse. What do you notice?

C

B D A


Page 10 of 12

Let’s see what else we can discover about _______ – _______ – _______ triangles. Complete the

chart with the lengths of the missing sides:

Once again, keep your answer in the simplest radical form.

Leg (a) Leg (b) Hypotenuse (c)

34

9

66

5

17

x

Work area

Notice anything?


Page 11 of 12

The

orem

7.8

45˚ – 45˚ – 90˚ Triangle Theorem

In a _______ – _______ – _______ triangle, the hypotenuse is _________ times as long as

either leg.

The ratios of the side lengths can be written:

________ – _________ – _________

The

orem

7.9

30˚ – 60˚ – 90˚ Triangle Theorem

In a _______ – _______ – _______ triangle, the hypotenuse is _________ as long as the

_______________ leg (opposite the __________ angle). The _______________ leg (opposite

the ________ angle) is _________ times as long as the shorter leg.

The ratios of the side lengths can be written:

_________ – _________ – _________

Practice 7.4

Page 461: 3 – 5, and more ….


Page 12 of 12

Page 461 – 2: 8 – 12, 23 – 25

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The Pythagorean Theorem & Special Right Triangles...The Pythagorean Theorem & Special Right Triangles We are all familiar with the Pythagorean Theorem and now we’ve explored one