A,B,C,D are coplanar as in the official diagram, with B and D on opposite sides of AC. ABC is equilateral; PAD is right isosceles at A; ∠PAC=π/2, ∠PCA=π/6, PA=1 and CD=2. Prove DA⊥PC.
Official paper · jm02-2024 · 1(a) · PDF 3
Official original and suggested answers ↗ · Suggested answer PDF page 8
Skills and prerequisite lessons
Working and explanation
BUILD THE REASONING
Hint 1
Find AC from right triangle PAC.
Hint 2
Prove DA perpendicular to two intersecting lines in plane APC.
Worked solution
Calculate AC and AD.
The converse of Pythagoras gives DA⊥AC.
Also DA⊥AP since PAD is right at A; therefore DA⊥plane APC, hence DA⊥PC.
DA⊥PC.
Checks and common pitfalls: Perpendicularity to just one line does not establish perpendicularity to a plane.
Reasoning checklist · self / teacher assessment
- Teaching assessment checklist, independently authored. Use the original paper for official marks.
- Calculate AC and AD.
- The converse of Pythagoras gives DA⊥AC.
- Also DA⊥AP since PAD is right at A; therefore DA⊥plane APC, hence DA⊥PC.
Think first. Reveal a hint when the class is ready.