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. Theorem 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 _______________________.
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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…
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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 ….
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
Page 4 of 12
Page 436 – 7: 8 – 10, 14 – 17
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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 _______________________________________________________
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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 ….
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
Page 7 of 12
Page 444: 9 – 23 ODDS
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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?
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
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 ….
Chapter 7: Trigonometry ................................................................................................. Sections 1 – 4
Page 12 of 12
Page 461 – 2: 8 – 12, 23 – 25
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