09 Solving triangles · g10-09-01 · 5–10 min
A triangle and three equal ratios
If one side changes scale, what happens to a/sinA?
Predict: Fix A=30°,B=60° and double side a. Predict all other sides and the three ratios.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Angle A (degrees) = 30; Angle B (degrees) = 60; Opposite side a = 3 | |
Case 2
Angle A (degrees) = 30; Angle B (degrees) = 60; Opposite side a = 6 | |
Case 3
Angle A (degrees) = 90; Angle B (degrees) = 90; Opposite side a = 3 | |
Revise: My claim needs the condition… because…
Exit: A=30°,B=90°,a=3. Find b.
09 Solving triangles · g10-09-02 · 5–10 min
Pythagoras with a correction term
What changes when the included angle is not90°?
Predict: Keep a=5,b=7 and compare c at C=60°,90°,120°.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Side a = 5; Side b = 7; Included C (degrees) = 60 | |
Case 2
Side a = 5; Side b = 7; Included C (degrees) = 90 | |
Case 3
Side a = 5; Side b = 7; Included C (degrees) = 180 | |
Revise: My claim needs the condition… because…
Exit: For a=5,b=7,C=60°, find c².
09 Solving triangles · g10-09-03 · 5–10 min
The same SSA data can make two triangles
How can two sides and a nonincluded angle give0,1 or2 solutions?
Predict: Fix A=30°,b=10. Predict the number of triangles at a=4,5,6,10.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Known A (degrees) = 30; Known side b = 10; Opposite side a = 4 | |
Case 2
Known A (degrees) = 30; Known side b = 10; Opposite side a = 5 | |
Case 3
Known A (degrees) = 30; Known side b = 10; Opposite side a = 6 | |
Revise: My claim needs the condition… because…
Exit: With A=30°,b=10, how many triangles exist at a=6?
09 Solving triangles · g10-09-04 · 5–10 min
The same two sides, different areas
Which included angle gives the greatest area for fixed sides?
Predict: For sides5 and7, compare areas at30°,90°,150°.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Side a = 5; Side b = 7; Included C (degrees) = 30 | |
Case 2
Side a = 5; Side b = 7; Included C (degrees) = 90 | |
Case 3
Side a = 5; Side b = 7; Included C (degrees) = 0 | |
Revise: My claim needs the condition… because…
Exit: Find the greatest area with sides4 and6.
09 Solving triangles · g10-09-05 · 5–10 min
Measure a distance without crossing the river
What can a baseline and two sighting angles determine?
Predict: Use a20m baseline and angles45°,60°. Predict whether the far point is above the baseline and whether its distance exceeds20m.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Baseline (m) = 20; First angle A (degrees) = 45; Second angle B (degrees) = 60 | |
Case 2
Baseline (m) = 20; First angle A (degrees) = 45; Second angle B (degrees) = 45 | |
Case 3
Baseline (m) = 20; First angle A (degrees) = 90; Second angle B (degrees) = 90 | |
Revise: My claim needs the condition… because…
Exit: For c=20,A=B=45°, find the perpendicular width.
