Geometer's Sketchpad
Java Sketchpad
Law of Sines - Ambiguous Case

Teachers Notes: Rationale/Level for Lesson

Objective:

In this activity students will use Geometer's Sketchpad, Cabri Geometry or the TI-92 to construct the SSA Ambiguous Case to demonstrate the possible number of triangles which can be constructed based on the length of the third side.

Activity:

1. Construction of the intial SSA:
   
   
   1. Construct point A
      
   2. Construct point B located horizontally to the right of A
      
   3. Select both points A and B and measure the distance between the points.-
      
   4. Move point B so that the length of AB is about 3.5 inches.
      
   5. Construct ray AB
      
   6. Construct point C
      
   7. Construct ray AC
      
   8. Select both points A and C and measure the distance between the points.
      
   9. Select Points C,A,and B. Measure the angle CAB.
      
   10. Select and move point C so that the length of AC is 3 inches and the measure of angle CAB is 30 degrees.
       
   
   
2. Constructing the third side:
   
   
   1. Construct point D in the interior of the constructed angle
      
   2. Construct segment CD.
      
   3. Select both points C and D and measure the distance between the points.
      
   4. Select points C,D, and A and measure angle CDA
      
   
   
3. Circle Construction:
   
   
   • Determine the number of possible solutions based on the length of the third side.
     
     
     1. Construct a circle with center C and passing through point D
        
     2. Select and move point D so that the circle intersects ray AD twice.
        
     3. Select and label the two points of intersection.
        
     4. Construct two radii from the center to each point of intersection. Measure the lengths. of the new line segments.
        
     
     Compare your construction to this drawing
     
   
   
4. Possible CASES for solutions>
   
   
   1. Select and move point D along ray AB so that the length of CD is about 1.5 inches
      
      
      1. Write down your observations for constructing a triangle based on the length of the third side.
         
      2. Write down the measure of angle CDA based on the placement of point D on ray AB.
         
      
      
   2. Select and move point D along ray AB so that the length of CD is about 2.5 inches
      
      
      1. Write down your observations for constructing a triangle based on the length of the third side.
         
      2. Write down the measure of angle CDA based on the placement of point D on ray AB.
         
      
      
   3. Select and move point D along ray AB so that the length of CD is about 3.5 inches
      
      
      1. Write down your observations for constructing a triangle based on the length of the third side.
         
      2. Write down the measure of angle CDA based on the placement of point D on ray AB.
         
      
      
   4. Select and move point D along ray AB so that the measure of angle CDA is 90 degrees.
      
      
      1. Write down your observations for constructing a triangle based on the length of the third side.
         
      2. Write down the length of CD based on the placement of point D on ray AB.
         
      
      
      
      

Conclusions:

1. Summarize the results above.
   
   
   1. How many different triangles could be constructed in each case?
      
   2. What can you determine about the lengths of the third side and the
      number of triangles that could be constructed?
      

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