Grade 11 · Triangle area and angle
Triangle area and angle: A nearly flat triangle
Triangle area and angle: investigate a nearly flat triangle with second side b = 4; included angle (degrees) = 165.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
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Understand what you are seeing
The idea behind the motion.
Two sides and their included angle determine a triangle's area. The perpendicular height is b sin θ, so the area is one half of the fixed base three times that height. Supplementary angles have equal sine and equal area, even though their opposite sides usually differ. This investigation starts with Second side b = 4; Included angle (degrees) = 165. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.
A relationship to keep
Area=(3b sin θ)/2; c²=9+b²−6b cos θ
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Set up the mathematical model
Keep the base fixed at three units and identify which angle is between the two known sides. The starting case is “A nearly flat triangle.”
- STEP 2
Follow the changing quantity
Rotate the second side from 15° toward the selected angle. The perpendicular height changes continuously while both side lengths remain fixed.
- STEP 3
Explain and test the result
Compare an acute angle with its supplementary obtuse angle. Use the cosine rule to explain why equal area does not determine the third side.
Your turn to explain
Make a prediction. Test your reasoning.
Keep Second side b = 4; Included angle (degrees) = 165. Pause the timeline at 80%. Given included angle = 135, calculate triangle area, opposite side length. Show the substitution into the displayed formula.
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
Height is 4 sin(135°)=2.828427. Hence area=3(2.828427)/2=4.242641; the cosine rule gives c=√(9+4²−6(4)cos(135°))=6.478469. Results: Triangle area: 4.243; Opposite side length: 6.478. Decimal values are rounded; retain the original parameters when checking.
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