MCR 3U0
Name:_____________
Lets Investigate Sine Law – The Ambiguous Case
Note:




The shortest distance from a point to a line is ________________________________________
In a triangle, the longest side is opposite the
angle.
Angle sum of a triangle is ______________________
In an oblique triangle:

Either all angles are acute
 Or one angle is
and the other two are
Given: A triangle that is constructed by the following information.
 A = 40o, AC = b = 10 cm, B = 90o
C
Step 1: Calculate the height of this triangle to base AB from the given information.
A
B
The height of this triangle is _______________________
This information is important in our investigation of any triangle given S.S.A.
Purpose: A partial triangle, ABC can be constructed with A = 40o, b = 10 cm. By varying the length of
side ‘a’ we will come to conclusions about:
a)
The number of triangles that can be constructed with S.S.A.
b)
Under what to expect each condition.
On the line segment given construct A = 40o, AC = 10 cm.
A
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Tools
protractor
long ruler
compass
MCR 3U0
Name:_____________
Data Recording:
Length of side a
Complete the following chart as stated to determine a conclusion regarding SSA.
Sketch with Number of Triangle(s) with Observations.
6.43 cm
4 cm
7 cm
10 cm
12 cm
The case of SSA with the sin law creates the following conclusions:
 height of triangle = b sinA

If  A is acute,  A  90o
Condition
(with respect to height)
side a, opposite from the given angle is
less than the height.
a  b sinA
side a, across from A is larger than the
height but is smaller than the side b, next
to the given angle.
b sin A  a  b
case 1: side a = height
a = b sin A
case 2: side a  side b
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Number of
Type of
Triangles
Possible
Example
A = 38o
b = 14 cm
a = 7 cm
 height =
o
A = 51
a=8
b = 10
 height =
o
A= 26
b = 9.1 cm
a = 4 cm
A = 61o
a = 22
b = 18
 height =
MCR 3U0
Name:_____________
* If  A is obtuse
 A  90o
 A  90o
if a  b
if a  b
no triangle (longest side is not opposite largest angle)
one obtuse triangle
Example:
A= 102o
a = 22
b = 18
Draw a rough sketch of the following triangles marking the given information. Determine which triangles have
no solution, one solution or two solutions. Redraw your triangles to reflect your solution.
a) In  ABC, a = 3 m, b = 15 m and A = 16o.
 Calculate the height to AB.
 How many distinct triangles can be drawn?
b) In  ABC,  A = 30o, a = 20 mm, b = 16 mm.
 Calculate the height to AB.
 How many distinct triangles can be drawn?
c) In ABC,  A = 40o, a = 22 cm, b = 27 cm.
 Calculate the height to AB.
 How many distinct triangles can be drawn?
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