Grade 11 · Triangle area and angle · 526 of 600
Triangle area and angle · Second side b=1.5; Included angle (degrees)=120
Use the included angle C=78°. Find the area and the side opposite C. Givens: Second side b=1.5; Included angle (degrees)=120.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Two sides of a triangle are 3 and 1.5 length units. The animation moves the included angle from 15° toward 120°. Use the included angle C=78°. Find the area and the side opposite C.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Area is one half the product of the two sides and sin C. The cosine rule determines the third side.
Work through the reasoning
Step 1
Identify the model and target
Two sides of a triangle are 3 and 1.5 length units. The animation moves the included angle from 15° toward 120°. Use the included angle C=78°. The governing relation is Area=(3b sin θ)/2; c²=9+b²−6b cos θ. Area is one half the product of the two sides and sin C. The cosine rule determines the third side.
Step 2
Substitute and calculate
Height is 1.5 sin(78°)=1.467221. Hence area=3(1.467221)/2=2.200832; the cosine rule gives c=√(9+1.5²−6(1.5)cos(78°))=3.062482.
Step 3
Check the mathematical meaning
The area cannot exceed 3(1.5)/2=2.25. The opposite side 3.062482 lies strictly between |3−1.5|=1.5 and 3+1.5=4.5. Included angle: 78; Triangle area: 2.201; Opposite side length: 3.062. Decimal values are rounded, so use unrounded intermediate values.
The answer
Height is 1.5 sin(78°)=1.467221. Hence area=3(1.467221)/2=2.200832; the cosine rule gives c=√(9+1.5²−6(1.5)cos(78°))=3.062482. The area cannot exceed 3(1.5)/2=2.25. The opposite side 3.062482 lies strictly between |3−1.5|=1.5 and 3+1.5=4.5. Animation check: Included angle: 78; Triangle area: 2.201; Opposite side length: 3.062. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
The mathematical idea
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. Two sides of a triangle are 3 and 1.5 length units. The animation moves the included angle from 15° toward 120°. Use the included angle C=78°. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
Area=(3b sin θ)/2; c²=9+b²−6b cos θ
03 · Reflect and transfer
Explain what changes and why.
Replace C by 180°−C. Why is the area unchanged while the opposite side generally changes?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.