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:

- Construction of the intial SSA:

- Construct point A

- Construct point B located horizontally to the right of A

- Select both points A and B and measure the distance between the points.-

- Move point B so that the length of AB is about 3.5 inches.

- Construct ray AB

- Construct point C

- Construct ray AC

- Select both points A and C and measure the distance between the points.

- Select Points C,A,and B. Measure the angle CAB.

- Select and move point C so that the length of AC is 3 inches and the measure of angle CAB is 30 degrees.

- Constructing the third side:

- Construct point D in the interior of the constructed angle

- Construct segment CD.

- Select both points C and D and measure the distance between the points.

- Select points C,D, and A and measure angle CDA

- Circle Construction:

- Determine the number of possible solutions based on the length of the third side.

- Construct a circle with center C and passing through point D

- Select and move point D so that the circle intersects ray AD twice.

- Select and label the two points of intersection.

- 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

- Possible CASES for solutions>

- Select and move point D along ray AB so that the length of CD is about 1.5 inches

- Write down your observations for constructing a triangle based on the length of the third side.

- Write down the measure of angle CDA based on the placement of point D on ray AB.

- Select and move point D along ray AB so that the length of CD is about 2.5 inches

- Write down your observations for constructing a triangle based on the length of the third side.

- Write down the measure of angle CDA based on the placement of point D on ray AB.

- Select and move point D along ray AB so that the length of CD is about 3.5 inches

- Write down your observations for constructing a triangle based on the length of the third side.

- Write down the measure of angle CDA based on the placement of point D on ray AB.

- Select and move point D along ray AB so that the measure of angle CDA is 90 degrees.

- Write down your observations for constructing a triangle based on the length of the third side.

- Write down the length of CD based on the placement of point D on ray AB.

Conclusions:

- Summarize the results above.

- How many different triangles could be constructed in each case?

- What can you determine about the lengths of the third side and the

number of triangles that could be constructed?

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