g10-09-05 · Syllabus supplement
Measure a distance without crossing the river
What can a baseline and two sighting angles determine?
1 MIN · PREDICT ON PAPER
Use a20m baseline and angles45°,60°. Predict whether the far point is above the baseline and whether its distance exceeds20m.
Discuss or write first. No typed answer is required to continue.
The sightline AP and perpendicular width are different measurements.
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
A baseline c with base angles A,B fixes C=180°−A−B. The sine rule gives AC=c sinB/sinC; its perpendicular river width is AC sinA.
Conditions: c>0,A>0,B>0,A+B<180°. Assume a straight baseline, a common horizontal plane and accurate sighting angles.
Mathematical justification: The constructed triangle obeys the sine rule; the right triangle formed by dropping the perpendicular gives width=AC sinA.
1 MIN · EXIT CHECK
For c=20,A=B=45°, find the perpendicular width.
Check after answering
10 metres; the far point is above the midpoint of the baseline.
Teacher guidance and extension
Before: Triangle angles, sine and cosine.
Ask: Which segment is the sightline and which one represents width?
Watch for: Calling the slant sightline the perpendicular river width.
Challenge: Try two angles whose sum approaches180° and explain why the inferred distance grows.
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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