## Thursday, March 3, 2011

### How to Do Determining the Number of Triangles in the Ambuguous case (SSA): A Contruction Exploration

The ambiguous case.
SSA

See the Exploration on Page 480 of the text as a guideline.  Due Thursday 3/10/11.

We wish to construct triangle ABC given angle A, side AB, and side BC.

Exploration 1
1. DRAW
a. 120 degree angle with a protractor.
b. 7 cm segment
c. 6 cm segment
d. 5 cm segment

2a.  Copy angle A.
Now CONSTRUCT triangle(s) ABC if you can.  See diagram 2a below.
angle A = 120 degrees
AB = 6
BC = 7

2b. Copy angle A.
Now CONSTRUCT triangle(s) ABC if you can.  See diagram 2b below.
angle A = 120 degrees
AB = 6
BC = 6

2c. Copy angle A.
Now CONSTRUCT triangle(s) ABC if you can.  See diagram 2c below.
angle A = 120 degrees
AB = 6
BC = 5

3. Describe what you did.

4. What did you discover about the number of triangles you can make for 2a, 2b, and 2c?

5. Write down the "rules"

Exploration 2
1. DRAW  30 degree angle with a protractor.

2. Copy angle A

3. DRAW ray AC  and ray AB  (see diagram 3 below)

4. CONSTRUCT a perpendicular ( also known as the altitude or h ) from AC to ray AB to form a right angle  (see diagram 6b below)

5. Measure h

6a.  Let BC < h.
Now CONSTRUCT triangle(s) ABC if you can.

6b.  Choose BC so that BC = h.
Now CONSTRUCT triangle(s) ABC if you can.

6c.  Choose BC so that BC > AC  > h.
Now CONSTRUCT triangle(s) ABC if you can.

7. Describe what you did.

8. What did you discover about the number of triangles you can make for 6a, 6b and 6c?

9. Write down the "rules"

Exploration 3
1. DRAW
a. 30 degree angle with a protractor.
b. 7 cm segment
c. 6 cm segment

2.  Now CONSTRUCT triangle(s) ABC if you can.  See the diagram 3 below.

3. CONSTRUCT the altitude h by constructing a perpendicular from point C to line AB

4. Measure h

5. Describe what you did.

6. What did you discover about the number of triangles you can make?

7. Write down the "rules"

Question 1
Explain why sin C is the same in both triangles in the ambiguous case (exploration 3).  This is why the Law of Sines is also ambiguous in this case.

Question 2
What did you discover about the number of triangles you can make given SSA (two sides and an angle)? Summarize the "rules" for all cases.

Go HERE after 3/11/11 to see if you got the "rules" correct!