g10-09-02 · Syllabus supplement
Pythagoras with a correction term
What changes when the included angle is not90°?
1 MIN · PREDICT ON PAPER
Keep a=5,b=7 and compare c at C=60°,90°,120°.
Discuss or write first. No typed answer is required to continue.
The correction term depends on the included angle.
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
The cosine rule is c²=a²+b²−2abcosC. The correction vanishes at90°, is negative for acute C and positive for obtuse C.
Conditions: a,b>0 and0°<C<180° for a nondegenerate triangle; C must be their included angle.
Mathematical justification: Place the sides at(0,0),(b,0),(acosC,asinC). Squaring the endpoint difference gives a²+b²−2abcosC.
1 MIN · EXIT CHECK
For a=5,b=7,C=60°, find c².
Check after answering
25+49−35=39.
Teacher guidance and extension
Before: Triangle angles, sine and cosine.
Ask: What is cos90°, and why does that restore Pythagoras?
Watch for: Using a nonincluded angle in the cosine-rule expression.
Challenge: Compare c with a+b as C approaches180°.
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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