Wednesday, March 16, 2011

How to do Determining the number of Triangles in the Ambiguous case (SSA): A construction exploration

Exploration 1a:

AC = 120 degrees
AB = 6 cm
BC = 7 cm

1) Draw a 120 degree angle, 7 cm, 6 cm, and 5 cm line (6 and 5 cm line also used for other two diagrams in exploration 1)
2) Copy angle A
3) Fix compass to length 6 cm. Place pointy end on A and pencil end on the intial side. This will be point B.
*Step 1-3 are also used to start the other two diagrams of exploration 1
4) Fix compass to 7 cm length. Place pointy end on B and make an arc that intersects the terminal side (ray AC) The intersection will be point C.
5) construct a perpendicular to form height.
Exploration 1b:

Exploration 1c:

1) Repeat Step 1-3 of first diagram of exploration 1
2) Fix compass to 5 cm length. Place pointy end on B and make an arc that intersects ray AC. The arc did not intersect AC, it intersected AB. Therefore creating a line, not a triangle.
Exploration 2a:
1)Draw 30 degree angle with a protractor
2) Copy 30 degree angle
3) Draw ray AC and AB
4) AC can be any length, in this case, I fixed the compass to length 7 cm
5) Construct a perpendicular from AC to ray AB to form a altitude
*These 5 steps are the same for the other two triangles in exploration 2

6) First Case, let BC be less then height, which eventually, it won't form a triangle, because if BC is less than h then BC won't touch ray AB (point B is nonexistence)
Exploration 2b:

Exploration 2C:

Exploration 3:

1) Draw a 30 degree angle, 7 cm line segment and 6 cm line segment
2) Copy the 30 degree angle
3) Fix compass to 7 cm length. Place pointy end of compass on angle A, then place pencil end somewhere along the terminal side of the angle. That point will be AC.
4) Fix compass to 6 cm length. Place pointy end on C, then place pencil end to the left of C and right of C. Make arcs. These will be the two B points.
5) Construct altitude by constructing a perpendicular bisector.
6) Measure h.

1) Because sin A remains the same in the ambiguous case, sin c also remain the same
2) If angle A is obtuse, then side BC (long side) has to be greater than side AB in order to form a triangle. A triangle cannot form if BC is less than or equal to AB.
In a right triangle, side BC has to equal the height of the triangle. BC cannot be less than h.
BC could also be greater than h, but BC also has to be greater than AC as well and angle A has to be acute. Though this is not a right triangle, one obtuse triangle and two acute

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