Download topic 14 right triangles

TOPIC 14
right triangles
Let’s begin with some facts that we have encountered with right triangles:
 There can be at most _________ right angle(s) in a right triangle (on a plane in
Euclidean space)
 The acute angles of a right triangle are ____________________________.
 The longest side of a right triangle is called the ________________________.
 The shorter two sides of a right triangle are called the ______________.
 The area of a right triangle is the half the product of a base and its height, which
becomes one-half the product of its _____________.
Make a sketch of a right triangle so that the hypotenuse is on the bottom of the drawing. Call it
ABC . Then drop a perpendicular from the angle at the top to the hypotenuse, and call that
altitude AD .
You have created three triangles, and they are all similar to each other. Mark one of the acute
angles of ABC , and mark any other angles in the drawing which must be congruent to that
angle. Mark the other acute angle in ABC ,and find all other angles which are congruent to it.
Topic 14: Right Triangles
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Then list the three similar triangles, one after another after another. Be careful to keep the
letters in their proper, corresponding order.
Finally, list the extended proportions for each pair of similar triangles, and we will decide which
of these is most important.
In any proportion
a  q , a and c are called the extremes, and p and q are called the means
p c
of the proportion. Something special is happening with the similar triangles which appear in the
right triangle above.
You will notice that some of the proportions have repeated one of the sides, so you get
something in this form:
a b
b c
The means are equal, so this situation is called a geometric mean. This says that “b is the
geometric mean between a and c.”
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Solve these proportions for the geometric means.
1.
4.
4b
b 6
2.
10  b
b 6
3.
5 b
b 20
Find the geometric mean between 12 and 20. (Set it up in a proportion, and solve for the
equal means.)
Find the geometric mean between the given pair of numbers.
5.
6 and 10
8.
h = _____
6.
8 and 6
7.
4 and 16
Topic 14: Right Triangles
9.
b = _____
10.
a = ______
11.
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12.
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Let’s revisit the proportions.

Write the three extended proportions which come from the similar triangles.

Keep the proportions (not the extended ones) that contain a geometric mean.

Look at the two of them, one which has an a2 and the one with a b2.

Cross multiple those two, and add them together. We will do a little bit of algebra … and
something memorable will appear.
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Solve for x in each of the right triangles.
13.
14.
15.
16.
17.
18.
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19.
20.
Pythagoras in Action
(Do it Right!)
Solve for the variable noted.
21.
22.
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23.
25.
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24.
AC = 20 , DB = 14.
AB = ______
26.
PS = ________
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27.
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28.
MO = 20
x = ______
x = ______
h = _____
Let’s think about the distance formula that we studied earlier.
The length of PQ is the length of
the hypotenuse of the triangle.
So (PQ)2 = (x2 – x1)2 + (y2 – y1)2
Then our distance formula becomes:
PQ = (x 2 - x1 ) 2 + (y 2 - y1 )2
See that you can find about the times or eras
when Pythagorean made his contributions, and
when Rene Descartes made his contributions to
coordinate geometry.
y = _____
c = ______
Topic 14: Right Triangles
Two More Proofs of the
Pythagorean Theorem
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The proof attributed to President Garfield
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Special Right Triangles
29.
Pythagorean triples
If all three sides of a right triangle are integers, then the set of side lengths is called a
Pythagorean triple.
List the hypotenuse of each of these right triangles:
a.
3 , 4 , _____
b.
5, 12, _____
c.
8, 15, _____
d.
7, 24, _____
List three multiples of each of the triples above. These are also Pythagorean triples.
30.
Cut a square into two congruent pieces with one of its diagonals. The triangles which
result have angles of 450, 450, and 900. So it is called a 45-45-90 triangle.
Calculate the exact value of the missing side of each of these.
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a.
b.
c.
d.
e.
f.
In general, what is the value of the hypotenuse if the legs of a 45-45-90 triangle (which is
isosceles) has value x?
Topic 14: Right Triangles
31.
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Cut an equilateral triangle in half by dropping an altitude. The two triangles each have
angles of 300, 600, and 900.
Because the two triangles are congruent (Why?), then the base is bisected by the altitude.
Thus, one leg is half as long as the hypotenuse.
Calculate the missing side of each of these.
a.
d.
L.
b.
e.
c.
f.
In general, what is the value of the longer leg if the legs of a 30-60-90 if the shorter leg has value
x?
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Apply the general patterns that you recorded above for these special right triangles.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
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44.
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AB = _______
BF = _______
45.
s=
perimeter =
46.
The side of a square has measure 4 2 . Its diagonal has measure __________.
47.
The side of a square has measure 6 7 Its diagonal has measure __________.
48.
The diagonal of a square has measure 9
2 . Its sides have measure __________.
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49.
The diagonal of a square has measure 18. Its sides have measure __________.
50.
The diagonal of a rectangle has measure 20 and one side is 10. The perimeter is
________________.
51.
Calculate x.
52.
Calculate x.
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Special Angles and trigonometry
Here are the special angles drawn on the unit circle (radius = 1). Keep this diagram for lots of
future reference. It is really helpful, especially if you are a visual thinker.
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Mark these 30-60-90 and45-45-90 triangle with a correct set of measurements. (You choose
which numbers you wish, as above.) Fill in the chart with exact values (no decimals):
 (degrees)
30
45
60
sin 
cos 
tan 
If you know these nine entries really well, then you know the entire chart below! Use the
ratios that appear on these special triangles with (1) what you know about reference angles, (2)
the signs that occur in each of the quadrants, and (3) how the x and y coordinates match with trig
ratios.
Here are the quadrant angles and their trigonometry ratios:
sin 
cos 
90 o
180 o
270 o
360 o
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Fill in the chart below.
 (degrees)
sin 
cos 
tan 
cot 
sec 
csc 
sin 
cos 
tan 
cot 
sec 
csc 
0
30
45
60
90
120
135
150
180
210
225
240
270
 (degrees)
300
315
330
360
Afterwards, you should be able to produce the exact values for each of these, eventually
without looking at the chart.
Example: You are asked what the cos 2400 is. Here is your thought process: 2400 is in Quad
1
1
III, and its reference angle is 60. The cos 60 = , so cos 240 = - ( because the cosine
2
2
matches with the x-coordinate, and the x-coordinates in Quad III are negative.)
Example: You are asked for the csc 330. Your thought process: Cosecant (“csc”) is the
reciprocal of sine. So think of this first as a sine problem. 3300 is in Quad IV, and its reference
1
is 300. Then sin 30 = . The reciprocal is csc 30 = 2. In Quad IV, the sine is matched with the
2
y=coordinate, and the y-coordinate is negative. So the answer is csc 330 = -2
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53.
sin 120
54.
cos 300
55.
tan 60
56.
tan 135
57.
sin 45
58.
sec 180
59.
csc 60
60.
csc 300
61.
sin 180
62.
sin 270
63.
cos 210
64.
tan 225
65.
cot 315
66.
cot 750
67.
sin 390
68.
cos 540
69.
sec 60
70.
sin 510
71.
cot (-210)
72.
sin (-30)
73.
sec (-150)
74.
cos (-270)
75.
sec (-135)
76.
cos (-180)
77.
tan 420
78.
cos 300
79.
csc 240