g10-09-04 · Syllabus supplement
The same two sides, different areas
Which included angle gives the greatest area for fixed sides?
1 MIN · PREDICT ON PAPER
For sides5 and7, compare areas at30°,90°,150°.
Discuss or write first. No typed answer is required to continue.
Area follows the altitude, not only the two side lengths.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Area is ab sinC/2. It is greatest at90°; supplementary included angles have the same area but generally different third sides.
Conditions: a,b>0 and0°<C<180°. Endpoint angles give zero area and a degenerate figure.
Mathematical justification: Drop the altitude h=a sinC onto side b, so area=bh/2=ab sinC/2. Since sinC≤1, the bound ab/2 is attained at90°.
1 MIN · EXIT CHECK
Find the greatest area with sides4 and6.
Check after answering
12, attained at a90° included angle.
Teacher guidance and extension
Before: Triangle angles, sine and cosine.
Ask: Which altitude changes while both given sides stay fixed?
Watch for: Assuming equal two-side data imply equal area regardless of the included angle.
Challenge: Find two different included angles with equal nonzero area.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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