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Solving Triangles
Lecture Notes
page 1
Sample Problems
1. Solve the triangle. a = 3, b = 7, c = 6
2. Consider the triangle with sides 7, 9, and 12 units long. Let , , and
Compute each of the following.
a) cos ( +
+ )
b) cos
+ cos
+ cos
denote the angles of the triangle.
c) the exact value of the area of the triangle.
4
= : Let D be the midpoint of side AC
5
a) Find the exact value of the cosine of angle CDB.
3. In triangle ABC,
= 90 and cos
b) Compute the exact value of the cosine of angle ADB.
4. Point D is on side AB of triangle ABC; with \ACD = \BCD = 60 , AC = 5; and BC = 15. Find the
length of line segment CD.
5. A triangle has sides of length a, b, and c; which are consecutive integers in increasing order, and cos
Find cos .
=
5
:
16
6. Suppose that ABC is a equilateral triangle with all
sides of length 1 unit. We extend side AB by 1 unit
beyond B to get P , we extend side BC by 1 unit beyond C to get Q and side AC beyond A to get R.
Compute the length of the sides in triangle P QR.
7. All three sides of triangle ABC are 4 units long.
Compute the exact value of the cosine of the angle
shaded on the picture.
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Last revised: December 13, 2013
Lecture Notes
Solving Triangles
page 2
8. Consider a square with sides 1 unit long. To the inside of each side, we draw an isosceles triangle with
its greatest angle, opposite the unit long base, measures 150 . Consider all vertices of these triangles
that are not on the square. If we connect these vertices, we obtain a square. Compute the exact value
of the area of this square.
9. A trapezoid’s parallel sides are 18 and 24 units long. Another side is 15 units long, and the angle formed
between this side and the longer parallel side is 74:5 . Compute the fourth side of the trapezoid and its
angles.
10. Let ABC be a triangle with sides a, b, and c. We
draw squares on all three sides. The vertices of the
squares that are not on the triangle form a hexagon.
Label the sides of the hexagon that are not on the
square by x, y, and z. Prove that x2 + y 2 + z 2 =
3 a2 + b2 + c2 .
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Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 3
Answers for Sample Problems
1.)
25: 208 7655 ,
p
2 13
3.) a)
13
96: 379 37 ,
p
2 13
13
b)
58: 411 8645
15
4.)
= 3:75
4
2.) a)
13
5.)
20
9.) 14: 591 units, 105: 5 , 97:84440 , 82: 1556
6.)
1
p
b)
p
c) 14 5
37
27
p
3 21
7.)
14
7 unit
8.) 2
p
3
10.) see solutions
Solutions of Sample Problems
1. Solve the triangle. a = 3, b = 7, c = 6
When using the law of cosines, we will …rst …nd the largest angle in the triangle. This way, when we next
apply the law of sines (the law of cosines is also an option), we know that we are solving for angles that
MUST be acute.
First we will use the law of cosines to …nd .
b2 = a2 + c2
2
= a +c
b2
a2 + c2 b2
32 + 62 72
=
=
=
2ac
2 3 6
2ac cos
cos
Then
1
= cos
We can now …nd
again.
2ac cos
2
1
9
1
96: 379 37
9
by using either the law of sines or the law of cosines.
c2 = a2 + b2
2ab cos
cos
2
We will use the law of cosines
2ab cos
2
= a +b
c2
a2 + b2 c2
32 + 72 62
11
=
=
=
2ab
2 3 7
21
11
58: 411 8645
21
The third angle can be easily found.
Thus
1
= cos
= 180
And so the answer is
( + )
25: 208 7655 ,
180
(96: 379 37 + 58: 411 8645 ) = 25: 208 7655
96: 379 37 , and
58: 411 8645 .
2. Consider the triangle with sides 7, 9, and 12 units long. Let , , and
Compute each of the following.
a) cos ( +
b) cos
+ ) = cos 180 =
+ cos
denote the angles of the triangle.
1
+ cos
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Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 4
Solution: Let us label the sides a = 7, b = 9, and c = 12. We …rst use the law of cosines to compute cos
a2 = b2 + c2
2
= b +c
a2
2
2
b +c
a2
92 + 122 72
22
=
=
=
2ac
2 9 12
27
2bc cos
cos
We compute
2bc cos
2
similarly.
b2 = a2 + c2
2ac cos
cos
2
2ac cos
2
= a +c
b2
72 + 122 92
2
a2 + c2 b2
=
=
=
2ac
2 7 12
3
We will again use the law of cosines, this time to compute .
c2 = a2 + b2
2ab cos
= a +b
c2
a2 + b2 c2
72 + 92 122
1
cos
=
=
=
2ab
2 7 9
9
22 2 1
37
Now cos + cos + cos =
+
=
27 3 9
27
c) the exact value of the area of the triangle.
1
Solution: We will use the formula A = ab sin . We …rst need to compute sin . We know that sin
2
positive because is an angle in a triangle, and so 0 < < 180 .
s
r
r
p
p
1 2
1
80
4 5
2
sin = 1 cos = 1
= 1
=
=
9
81
81
9
2ab cos
2
2
is
Now we can compute the exact value of the area of the triangle.
p !
p
1
1
4 5
A = ab sin =
7 9
= 14 5
2
2
9
4
= : Let D be the midpoint of side AC
5
a) Find the exact value of the cosine of angle CDB.
3. In triangle ABC,
= 90 and cos
Solution: It is part of the problem to come up with a
picture that depicts the data. Sometimes that is not
easy and takes several attempts. However, a good
picture is often an essential part of solving a geometry
problem.
4
From the fact that this is right triangle and cos = , we conclude that this triangle is similar to the one
5
with sides 3, 4, and 5 units and label the sides as 3x, 4x, and 5x where x is a positive number. Since D is
the midpoint of line segment AC, both AD and DC are of length 2x.
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Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 5
We compute the length of line segment BD using the Pythagorean Theorem:
q
p
p
p
BD = (2x)2 + (3x)2 = 4x2 + 9x2 = 13x2 = 13x
We are now ready to …nd the cosine of angle CBD
p
2x
2
2 13
cos (\CDB) = p
=p =
13
13x
13
b) Compute the exact value of the cosine of angle ADB.
Solution 1: Angles CDB and ADB are supplements, they add up to 180 . For all angles ,
cos (180
)=
cos
and so
cos (\ADB) =
cos (\ADB) =
p
2 13
13
Solution 2: Consider triangle ABD.
We can compute the cosine of
= \ADB by stating the law of cosines for triangle ABD.
(AB)2 = (AD)2 + (DB)2 2 (AD) (DB) cos
p
p
2
13x
2 (2x)
13x cos
(5x)2 = (2x)2 +
p
25x2 = 4x2 + 13x2 4 13x2 cos
p
25x2 =
17 4 13 cos x2
p
25 = 17 4 13 cos
p
8 =
4 13 cos
8
p
= cos
4 13
p
2
2 13
p =
cos
=
13
13
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divide by x2
Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 6
4. Point D is on side AB of triangle ABC; with \ACD = \BCD = 60 , AC = 5; and BC = 15. Find the
length of line segment CD.
Solution: It is part of the problem to come up with a
picture that depicts the data. Sometimes that is not
easy and takes several attempts. However, a good
picture is often an essential part of solving a geometry
problem.
1
Let us denote line segment CD by x. Recall the formula A = ab sin .
2
expressing the area of this triangle in two di¤erent ways.
ATriangle ABC
1
(5) (15) sin 120
2
75 sin 120
75 sin 120
p
3
75
2
= ATriangle ADC + ATriangle BCD
1
1
=
(5) (x) sin 60 + (15) (x) sin 60
2
2
= 5x sin 60 + 15x sin 60
We will solve this problem by
multiply by 2
= 20x sin 60!
p
3
= 20x
2
divide by
p
3
2
75 = 20x
75
15
x =
=
= 3:75
20
4
5. A triangle has sides of length a, b, and c; which are consecutive integers in increasing order, and cos
Find cos .
=
5
:
16
Solution: First we would want to express the fact that the three sides are consecutive integers. The usual
labeling, a = x, b = x + 1, and c = x + 2 would work, but we will chose a smarter labeling that will cut down
on the computations, that is a = x 1, b = x, and c = x + 1. We will need to remember that x must be a
positive integer. Since cos is given, we will state the law of cosines involving cos :
c2 = a2 + b2
(x + 1)2 = (x
2ab cos
1)2 + x2
x2 + 2x + 1 = x2
2 (x
2x + 1 + x2
32x = 8x2
5 2
x
x
8
5 x2 x
32x = 8x2
5x2 + 5x
4x = x2
1) x
10 2
x
16
5
16
x
multiply by 8
27x = 3x2
0 = 3x2
0 = 3x (x
27x
9)
=)
x1 = 0; x2 = 9
Since x can not be zero, x = 9 is the only solution. Because we labeled b by x, this means that the triangle’s
c copyright Hidegkuti, 2013
Last revised: December 13, 2013
Solving Triangles
Lecture Notes
three sides are a = 8, b = 9, and c = 10 units. We will compute cos
a2 = b2 + c2
2
page 7
using the law of cosines again:
2bc cos
= b +c
a2
b2 + c2 a2
92 + 102 82
13
=
=
=
2bc
2 9 10
20
2bc cos
cos
2
6. Suppose that ABC is a equilateral triangle with all
sides of length 1 unit. We extend side AB by 1 unit
beyond B to get P , we extend side BC by 1 unit beyond C to get Q and side AC beyond A to get R.
Compute the length of the sides in triangle P QR.
Solution: By symmetry, triangle P QR is also equilateral and so we just need to compute the length of any
side. Consider triangle P BQ.
Side BP is 1 unit long and side BQ is 2 units long. Angle P BQ has measure 120 since it is the supplement
of the inner angle which measures 60 . We denote side P Q by x and state the law of cosines on triangle
P BQ.
x2 = 12 + 22
2 1 2 cos 120
x2 = 5
1
2
4
cos 120 =
1
2
x2 = 5 + 2
x2 = 7
x =
p
7
Since x is a distance, we discard the negative solution of the equation. The triangle P QR has sides of length
p
7.
c copyright Hidegkuti, 2013
Last revised: December 13, 2013
Lecture Notes
Solving Triangles
page 8
7. All three sides of triangle ABC are 4 units long. Compute the exact value of the cosine of the angle shaded
on the picture below.
Solution: We will solve this problem by stating the law of cosines for triangle CDE. We …rst need to …nd
the length of its three sides.
First, side EC is clearly 1 unit long. We can compute the length of CD via the Pythagorean theorem
CD2 + 22 = 42
CD2 = 12
=)
CD =
p
12
We will now use the law of cosines in triangle ADE to compute the length of ED.
ED2 = 22 + 32
2 2 3 cos 60
p
1
ED2 = 4 + 9 12
= 13 6 = 7 =) ED = 7
2
p
p
So now we know that EC = 1, CD = 12, and ED = 7. We need to compute the cosine of the angle
oposite side EC. Let us enote this angle by .
p
p 2
p p
2
12 =
12 +
7
2 12 7 cos
p
1 = 19 2 84 cos
p
2 84 cos
= 18
p
p
p
p
p
3 84
3 2 21
3 21
18
9
9 84
9 84
p =p =
=
=
=
cos
=
=
84
28
28
14
84
2 84
84
8. Consider a square with sides 1 unit long. To the inside of each side, we draw an isosceles triangle with its
greatest angle, opposite the unit long base, measures 150 . Consider all vertices of these triangles that are
not on the square. If we connect these vertices, we obtain a square. Compute the exact value of the area
of this square.
c copyright Hidegkuti, 2013
Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 9
Solution: This problem can be solved without the law of cosines, but it is much easier using it. Let us …rst
connect the four points, denoted by E, F , G, and H as shown on the picture below.
Consider …rst triangle ABF . By symmetry, AF = BF and so the triangle is isosceles. Since angle
AF B = 150 , the angles F AB and ABF must measure 15 . By symmetry, angle EAD must also measure
15 .
Consider now angle EAF . It must measure 90 (15 + 15 ) = 60 . Consider triangle AEF . By symmetry,
AE = EF and so the angles opposite those sides are also the same. Since the third angle, \EAF = 60 , the
other two angles must add up to 120 : Because they are equal, they must measure 60 and so the triangle
is equilateral and AE = EF = AF .
The area of the square is x2 and we can …nd its value by stating the law of cosines on triangle ABF .
x2 + x2
2 (x) (x) cos 150 = 12
p !
3
2
2
2x
2x
= 1
2
p
x2 2 + 3
= 1
x2 =
1
p
2+ 3
We rationalize our value for x2 .
1
1
2
p =
p
x =
2+ 3
2+ 3 2
2
p
p
3
2
3
p =
=2
1
3
p
3
This is the area of the square.
c copyright Hidegkuti, 2013
Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 10
9. A trapezoid’s parallel sides are 18 and 24 units long. Another side is 15 units long, and the angle formed
between this side and the longer parallel side is 74:5 . Compute the fourth side of the trapezoid and its
angles.
Solution: We will …rst prove a fact that is true for all trapezoids. The two angles formed at a trapezoid’s
side conntecting parallel sides are supplements, they add up to 180 .
proof: Consider the picture below. We denoted the angle by A by
extended line AD beyond D and line BC beyond C.
and the angle by B by . Then we
These new angles formed are again and because sides AB and CD are parallel. Now it is easy to see
that + \ADC = 180 and + \BCD = 180 .
The solution of a geometry problem is often very easy after a well chosen line was drawn in. Consider the
picture below.
Let us draw a line through C that is parallel to side AC. Now AECD is a parallelogram, because it has two
pairs of parallel sides. It is a propoerty of parallelograms that opposite sides have the same length. Thus
AE = 18 and consequently, EB = 24 18 = 6 units. Also, angle CEB = 74:5 because AD and CE are
parallel.
Our smart line reduced the problem considerably. We can use the law of cosines to compute side BC:
BC 2 = 152 + 62
BC
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2
2 15 6 cos 74:5
212: 897 092 306
=) BC
14: 590994 unit
Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 11
We can use the law of sines again to compute angle EBC (denoted by )
152 = 62 + BC 2
12BC cos
cos
2
2 6 BC cos
2
= 6 + BC
152
62 + BC 2 152
62 + (14: 590994)2 152
=
12BC
12 (14: 590994)
1
cos (0:1364832) 82: 1556
0:1364832
The other two angles can be found easily by using the fact that the angles along a side connecting parallel
sides are supplemental.
\ADC = 180
74:5 = 105: 5
and \BCD = 180
= 180
82: 1556 = 97: 8444
Thus the missing side and angles are: 14: 591 units, 105: 5 , 97:84440 , and 82: 1556 .
10. Let ABC be a triangle with sides a, b, and c. We draw squares on all three sides. The vertices of the
squares that are not on the triangle form a hexagon. Label the sides of the hexagon that are not on the
square by x, y, and z. Prove that x2 + y 2 + z 2 = 3 a2 + b2 + c2 .
proof: Consider the picture below.
We …rst establish that the angles, shown on the picture are complements to , , and , correspondingly.
Also, recall that for all angles , cos (180
) = cos . Let us state the law of cosines for the triangles
that include x, y, and z as a side.
x2 = a2 + c2 2ac cos (180
x2 = a2 + c2 + 2ac cos
y 2 = a2 + b2 2ab cos (180
y 2 = a2 + b2 + 2ab cos
)
)
and
z 2 = b2 + c2
z
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2
2
2bc cos (180
)
2
= b + c + 2bc cos
Last revised: December 13, 2013
Solving Triangles
Lecture Notes
page 12
We add these three equations and get
x2 + y 2 + z 2 = a2 + c2 + 2ac cos
+ a2 + b2 + 2ab cos + b2 + c2 + 2bc cos
x2 + y 2 + z 2 = 2a2 + 2b2 + 2c2 + 2ac cos
+ 2ab cos + 2bc cos
We will now "get rid" of the expressions 2ac cos , 2ab cos , and 2ab cos
on the triangle abc.
a2 = b2 + c2 2bc cos
2bc cos = b2 + c2 a2
b2 = a2 + c2
2ac cos = a2 + c2
2ac cos
b2
by using the law of cosines stated
c2 = a2 + b2 2ab cos
2ab cos = a2 + b2 c2
We substitute these into our equation
x2 + y 2 + z 2 = 2a2 + 2b2 + 2c2 + 2ac cos
2
2
2
2
+ 2ab cos + 2bc cos
2
= 2a + 2b + 2c + a + c
b2 + a2 + b2
c2 + b2 + c2
a2
= 3a2 + 3b2 + 3c2
This completes our proof.
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c copyright Hidegkuti, 2013
Last revised: December 13, 2013