Download Mth 97 Winter 2013 Sections 6.3 and 6.4 Section 6.3 – Applications

Mth 97
Winter 2013
Sections 6.3 and 6.4
A
Section 6.3 – Applications of Similarity
Name three pairs of similar triangles.
D
A
60°
30°
0
C
B
Results for any Right Triangle
D
(1) The height BD to the hypotenuse is the
geometric mean of AD and DC.
C
B
ABC is a right triangle. CD is the altitude of the
triangle from vertex C (also called the height).
AD CD

orCD  ( AD)( DB)
.
CD DB
(2) The three triangles formed are all similar to
each other. ABC  ACD  CBD
R
Using the Mean Proportional
Name the three similar triangles.
If UT= 5 and US = 8, find AC.
T
U
S
Side Splitting Theorem
B
DE AC
D
A
You have overlapping triangles so that the bases
are parallel, and the triangles share an angle. By
AAA Similarity, we know ABC  DBE. Another
E
useful result is that DE cuts the two sides of the
large triangle into proportional segments.
C
Conclude:
If _________________________________
AD CE

DB EB
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Mth 97
Winter 2013
Sections 6.3 and 6.4
Using the Side Splitting Theorem
C
B
If AB = 5, BC = 2, and AE = 7, find ED.
A
If AC = 10, AE = 8, and ED = 4, find BC.
E
D
If BE = 7, DC = 10, and AE = 2 + ED, find AE.
D
If DJ = 5, JI = 3, DE = 8, and FG = 7, find IH and EF.
J
E
I
F
H
G
Midsegment Theorem
B
In the special case where D is the midpoint of AB ,
DE AC
D
and E is the midpoint of CB , the lengths of the
bases of the two triangles have a special
relationship. DE is half of AC.
E
C
A
If _____________________________________
Conclude: DE  12 AC
Using the Midsegment Theorem
A
E
D
If DC = 16, find EB.
B
C
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Mth 97
Winter 2013
Sections 6.3 and 6.4
Corollary 6.12 – Midquad Theorem
If ABCD is any quadrilateral and E, F, G, H are midpoints as shown then EFGH is a parallelogram.
A
F
E
B
D
H
G
C
Section 6.4 – Using Right Triangle Trigonometry to Solve Geometry Problems
A
Trigonometry is the study of measures of ____________________.
Definitions of Trig Ratios (for the acute angles in a right triangle)
b
sine(acute angle) = length of opposite leg
length of hypotenuse
Sin A =
cosine(acute angle) = length of adjacent leg
length of hypotenuse
Cos A =
Cos B =
tangent(acute angle) = length of opposite leg
length of adjacent leg
Tan A =
Tan B =
A way to remember these is:
SOH
Sin B =
c
C
B
a
CAH
TOA
Compute the following. Give answers rounded to 2 decimal places.
cos A, if b = 7 and c = 12
sin B, if a = 5 and c = 12
tan B, if a = 5 and c = 13
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Mth 97
Winter 2013
Sections 6.3 and 6.4
Using a calculator to find the sine, cosine, and tangent ratios of angles
To find tan 43° (check that your calculator is set in degrees, not radians)
Press Tan 43 ENTER. Your result should be 0.9325150861 which rounds to 0.9325 (4 decimal places).
sin 43° =
cos 43° =
Compute the following. Give answers rounded to 2 decimal places.
Find b, if mA  27.3 and c = 5.
Find c, if mB  42 and b = 10.
At a horizontal distance of 150 feet from the base of a building, the line of sight to the top of the
building makes an angle of 21°with level ground. That means the angle of elevation to the top of the
building is 21°. How tall is the building to the nearest hundredth of a foot? Assume the building is
perpendicular to the ground.
Use Inverse trigonometric functions to find the approximate measure of an acute angle in a right triangle
If you know a trig ratio in a right triangle, you can find the measures of both acute angles by using sin-1,
cos-1, or tan-1. We usually round to tenths of a degree.
If tan A = 0.8391, enter tan-1 (2nd tan) .8391
_________________
mA 
If cos A = 0.9063, enter cos-1 (2nd cos) .9063
_________________
mA 
If sin A = 0.3581, enter sin-1 (2nd sin) .3581
_________________
mA 
Given a right triangle with right angle C with sides
measuring 3, 4, and 5, find the measure of angles A and B.
B
C
A
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Mth 97
Winter 2013
Sections 6.3 and 6.4
Theorem 6.13 – In a right triangle with acute angle A, sin A divided by cos A is tan A.
sin A
 tan A
cos A
Verify this theorem using the right triangle below.
5A
sin A =
cos A =
tan A =
13
5
C
12
B
sin A

cos A
Theorem 6.14 – In a right triangle with acute angle A, the sum of sin2A and cos2A is 1.
sin 2 A  cos 2 A  1
This theorem is verified for a right triangle on page 335.
Given that sin A = 0.6428, find the cos A.
Solving Right Triangles
To solve a right triangle means to find all the missing sides and angles.
B
12”
To find mA
To find mB
To find b use tan
or use Pythagorean Theorem
4”
C
b
We know mC  90
A
a=4
c = 12
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Mth 97
Winter 2013
Sections 6.3 and 6.4
Solve each right triangle.
E
20.43 m
8 cm
8.72 m
F
G
A
Given mG  45
B
Given a = 8.72
mF  90
b = 20.43
mC  90
EF = 8
Find mE 
Find c =
FG =
mA 
EG =
mB 
B
A
C
∆ABC is not a right triangle. Determine the area of ∆ABC given that
mA  78 , AB = 8 mm and AC = 17 mm. Hint: First draw a
perpendicular segment from angle B to segment AC to find the
triangle’s height (altitude).
C
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