Geometry Semester 2
Homework on the Laws of Sines and Cosines
1.
Name: _______________________________
Use the Law of Sines to find side BC. Round your answer to the nearest hundredth of a cm.
B
A
2.
40o
25o
C
12 cm
When two sides and an angle not between those two sides of a triangle are given, the triangle is
not always unique. Follow these steps to investigate what is happening in such a case.
a.
Suppose triangle ABC is a right triangle with angle C = 90o, angle A measures 25o, and
AB = 12. Then how long is BC?
B
12
25o
A
b.
25o
C
Now suppose angle C is not 90o, but BC is 6 units long. Then there are two possible such
triangles. What are the two possible measures of angle C?
B
12
6
25o
A
c.
6
C
C
Finally, suppose BC = 15. Then there is only one possible triangle. What is the measure
of angle C in that case?
B
15
12
A
25
o
C
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Problems 3-5:
Use the Law of Sines or the Law of Cosines to find x in each of the triangles.
Round your answers to the nearest hundredth of an inch.
3.
110
15 in
o
10 in
x
4.
123
o
15 in
23
o
x
5.
27
1 ft
o
5 in
x
Problems 6-8: Use the Law of Sines or the Law of Cosines to find θ in each of the triangles. Round
your answers to the nearest tenth of a degree.
θ
6.
13
6
63
o
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7.
18
10
θ
23
8.
Find θ to the nearest degree if it is an obtuse angle:
θ
14
35
o
20
9.
Find the value of x if cos  
1
:
3
x
x
θ
7
A
10.
In the quadrilateral at the right,
AC is perpendicular to AB. Find
the measure of P to the nearest
tenth of a degree.
3
B
4
6
C
5
P
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11.
Find the area of this triangle:
10 cm
40
o
12 cm
12.
Use Heron's Formula to find the area of this triangle:
7 cm
8 cm
9 cm
The remaining problems are designed to help you understand Heron's formula for the area of a
triangle.
13.
In the given rectangle ABCD, angles a, b, and g add to 90o.
a. What is tan a as a fraction in lowest terms?
15
B
E

b. How long is AE?
C
9
F
20
c. How long is EC?

d. How long is EF?

A
e. How long is FD?
e. What is tan b as a fraction in lowest terms?
f. What is tan g as a fraction in lowest terms?
g. Simplify: (tan )(tan ) + (tan )(tan ) + (tan )(tan ).
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
D
14.
In the given rectangle ABCD, angles a, b, and g add to 90o, and AB = CD= 1.
a. Since AB = 1, BE = tan a. What trig
E
B

C
function of a is AE?
1
F
EF
b. Since
 tan  , EF  AE tan  . Using
AE
your answer to part a, how long is EF in

A


terms of angles  and ?
EC
 cos  , EC  EF cos  . Using your answer to part b and the fact that
EF
1
sec  
, give EC in terms of .
cos 
c. Since
CF
 tan  , CF  EC tan  . Using your answer to part c, express CF in terms of
EC
 and .
d. Since
FD
 tan  , FD  AD tan  . Using the facts that AD = BC = BE + EC, BE = tan a,
AD
and your answer to part c, express FD in terms of ,  and .
e. Since
e. Using the fact that CD = AB = 1, simplify (tan )(tan ) + (tan )(tan ) + (tan )(tan ).
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D
15.
B
Suppose the sides of ABC measure
13, 14, and 15 units.
14
13
A
C
15
B
z
Let P be the incenter, and D, E, F
D
F
the points where PD  BC ,
y
PE  AC , and PF  AB, and let
P
x = AE, y = CD, z = BF.
A
x
C
E
B
a.
z
z
Then AF = x, CD = y, and BD = z.
D
F
Why?
x
A
y
P
x
E
C
y
b.
What is the value of 2x + 2y + 2z?
c.
Let s = x + y + z. Then what is the value of s?
d.
What are the values of x, y and z? (Hint: s = x + y + z = AC + z, so z = s – AC)
e.
Let   PAE  PCD,   PCE  PCD, and
B
z
  PBF  PBD . Express tan  , tan  , and tan  in
terms of r.
F
x
A
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 
z
D
y
P




x
E
y
C
f.
Since 2  2  2  180,       90. Use the following fact to solve for r:
Fact: If       90, then ( tan  tan  )  ( tan  tan  )  ( tan  tan  )  1
g.
The area of ABC can be found by adding the areas of 6 triangles, PAE, PAF ,
PBD , PBF , PCD, and PCE. Find the area of ABC that way.
B
Now we will prove Heron's formula in the general case:
z
Let the sides of ABC measure a, b, and c units, let P be
 
F
z
D
the incenter with PD, PE and PF radii of the incircle, let
x
x, y and z be the lengths of segments from the vertices of
ABC to D, E and F as shown, and let and  be the
A
y
P




measures of the bisectors of angles A, B and C. Let s be
x
E
y
the semiperimeter of ABC . Notice that a = y + z = s – x, b = x + z = s – y, and c = x + y = s – z.
16.
a.
Express tan  , tan  , and tan  in terms of r, x, y, and z.
b.
The area of ABC is the sum of the areas of the 6 triangles, PAE, PAF ,
PBD , PBF , PCD, and PCE. . Express the area of ABC in terms of r and s.
(Hint: Express it in terms of r, x, y, and z, and then factor out r.)
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C
c.
Use the fact that ( tan  tan  )  ( tan  tan  )  ( tan  tan  )  1 to solve for r in terms of
s, x, y, and z.
d.
Substitute the expression for r you found in part c into the formula for the area you found
in part b, and simplify. This should give you a formula for the area that depends only on s,
x, y, and z.
e.
Now substitute s – a for x, s – b for y, and s – c for z in the area formula you found in part d.
This should be Heron's Formula.
Page 8 of 8