The Ambiguous Case
Objective:
To determine the number of triangles possible when given two
sides and an angle opposite one of the sides.
Materials needed:
Each pair of students needs
two pieces of cardboard (5" x 8")
protractor
two pieces of string approximately 12" each
two thumb tacks
Instructions to the teacher for conducting the activity:
1. Students should be divided into groups of two prior to class.
2. One student should work with the materials, and the other student record
the results on the work sheet provided with the activity.
3. After the students have completed the work sheet, the teacher should
develop the ambigious case with the aid of the students and the results
of the activity.
Directions to the student:
1. Place the pieces of cardboard so that the length is 8"
and the width is 5".
2. Draw a horizontal line (approximately 7" long) on each board about
1" from the bottom.
3. On one piece of cardboard, place a point C on the reference line
near the left edge. Use the protractor to draw an angle, in standard position,
with a measure of 30 degrees and vertex C. Write 30 degrees in the
interior of the angle. Write A to label the point on the terminal
side that is 4" from the vertex.
4. On the other piece of cardboard, choose a point D about 2.5"
from the right end of the reference line. With a protractor, draw an angle,
in standard position, with a measure of 150 degrees and vertex D.
Use B to label the point on the terminal side that is 4" from
the vertex.
5. Tie a knot in one end of each piece of string. Place a thumb tack
through the knot in each string. Place one thumb tack at point A
and the other thumb tack at point B.
6. Complete "The Ambigious Case" work sheet.
Extension:
Students can generalize the results to determine if a triangle
exists, or how many exist, when they are given two sides and an angle opposite
one of the sides.
The Ambigious Case
Worksheet
Answer the following questions using A as the center of a circle
and a portion of the string as a radius. NOTE: The radius is actually
"side c" (opposite angle C) of a triangle.
For each of the following radii, in how many places will the circle cross
the reference line?
1. r = 1.5"
2. r = 2"
3. r = 3 "
4. r = 4"
5. r = 4.5"
6. For each question above, how many triangles will be formed using angle
C and side AC?
7. In #2, what kind of triangle is formed?
8. What kind of triangle resulted in #4?
Answer the following questions using B as the center of a cirlce
and a portion of the string as a radius. NOTE: The radius is actually
"side d" (opposite angle D) of a triangle.
For each of the following radii, in how many places will the circle cross
the reference line?
1. r = 1.5"
2. r = 2"
3. r = 3"
4. r = 4"
5. r = 4.5"
6. For each question above, how many triangles will be formed using angle
D and side BD?