Transcript ambiguous-case-triangles

Ambiguous Case Triangles
Can given numbers make a triangle? Can
a different triangle be formed with the
same information?
Conditions for Unique
Triangles
ASA
two angles must sum to less than 180º
SSS
AAS
two angles must sum to less than 180º
SAS
two shortest sides are longer than the third side
Any set of data that fits these conditions will result in one unique triangle.
Ambiguous Triangle Case
(aka the ‘bad’ word)
b
a
A
This diagram is deceiving -- side-side-angle
data may result in two different triangles.
SSA
Side a is given but it might be possible to ‘swing’ it to either
of two positions depending on the other given values.
An acute or an obtuse triangle may be possible.
Determining the Number of
Possible Triangles
The Textbook Method
3.4 The Ambiguous Case Of The Law Of Sines
If two sides and one angle opposite to one of them are given, there can be complications in
solving triangles. In fact, from the given data, we may determine more than one triangle or
perhaps no triangles at all.
Let the angle be C and sides be c and b of triangle ABC given then
Here, R.H.S. is completely known and hence Ð B can be found out.
Also Ð C is given \ ÐA = 1800 - (B + C). Thus triangle ABC is solved.
But when sin B has R. H.S. with such values that the triangle can't be com pleted.
Observe,
(I) When Ð C < 900 (acute angle)
(a) If c < b sin C then 1 i.e. sin B > 1 which is impossible (as sine c ratio never exceeds 1)
(b) c = b sin C then = 1 i.e. sin B = 1 \ Ð B = 900 i.e triangle ABC is c right triangle.
(c) If c > b sin C then < 1 i.e. sin B < 1. But within the range 00 to 1800 ; there are two values of
angles for a sine ratio as sin q = sin (180 - q), of which one is acute and the other is an obtuse.
e.g. sin 300 = sin 1500
But both of these may not be always admissible.
If ÐB < 900 (acute), the triangle ABC is possible as c > b then Ð C > Ð B. But if Ð B > 900
(obtuse) then Ð C will be also obtuse ; this is impossible as there can't be two obtuse angles in a
triangle.
If c < b and Ð C is acute then both values of B are admissible . Naturally there will be two values
of Ð A and hence two values of a. Hence there are two triangle possible.
(II) When Ð C < 900 (obtuse angle).
Here (1) If c < b then Ð C < Ð B but then B will also become an obtuse angle. It is also
impossible.
(2) If c = b then Ð B = Ð C. Hence Ð C is obtuse makes Ð B obtuse too which again impossible.
(3) If c > b then Ð C > Ð B. But again there are two possibilities (i) Ð B is acute (ii) and Ð B is
obtuse.
If Ð B is acute the triangle is possible and when Ð B is obtuse the triangle is impossible. For the
obtained values of the elements when there is ambiguity (i.e. we are unable to draw such a
triangle) to determine the triangle ; it is called an 'ambiguous case'.
The easy way
1) Use Law of Sines
a
b

sin A sin B
2) Sum of the angles
in a triangle = 180º
Example (2 triangles)
Given information
mA = 17º
a = 5.8
b = 14.3
Set up Law of Sines
5.8
14.3

sin 17 sin B
Find mB in quadrant I
mB  46º
Solve for sin B
14.3 sin 17
sin B 
5 .8
Find mB in quadrant II
mB  180 – 46 = 134º
Find mC
Find mC
mC  (180 – 17 – 46)  117º
mC  (180 – 17 – 134)  29º
Both values of C are possible, so 2 triangles are possible
Example (1 triangle)
Given information
Set up Law of Sines
mA = 58º
a = 20
b = 10
20
10

sin 58 sin B
Find mB in quadrant I
mB  25º
Solve for sin B
10 sin 58
sin B 
20
Find mB in quadrant II
mB  180 – 25 = 155º
Find mC
Find mC
mC  (180 – 58 – 25)  97º
mC  (180 – 58 – 155)  -33º
Only one value of C is possible, so only 1 triangle is possible
Example (0 triangles)
Given information
mA = 71º
a = 12
b = 17
Set up Law of Sines
Solve for sin B
12
17

sin 71 sin B
17 sin 71
sin B 
12
sin B  1.3395
No value of B is possible, so no triangles are possible
Law of Sines Method
1) Use Law of Sines to find angle B
-If there is no value of B (for example, sin B = 2),
then there are no triangles
Remember, sin x is positive in both quadrant I and II
2) Determine value of B in quadrant II
(i.e. 180 – quadrant I value)
3) Figure out the missing angle C for both values of
angle B by subtracting angles A and B from 180
4) If it is possible to find angle C for
-both values of B, then there are 2 triangles
-only the quadrant I value of B, then
only 1 triangle is possible