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. Find the volume of PBCD.

01 / Standard#Your turn

Official paper · jm02-2024 · 1(b) · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

PA is perpendicular to the base plane because it is perpendicular to AC and AD.

Hint 2

Add the adjacent angles DCA and ACB.

Worked solution

The base triangle is right at C.
$\sin\angle DCA=1/2\implies\angle DCA=\pi/6,\quad\angle DCB=\pi/6+\pi/3=\pi/2$
Use base area and perpendicular height.
$[BCD]=\frac12(2)(\sqrt3)=\sqrt3,\quad V=\frac13[BCD]\,PA=\frac{\sqrt3}3$

Volume √3/3.

Checks and common pitfalls: The relative positions in the diagram determine that the two angles add.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The base triangle is right at C.
$\sin\angle DCA=1/2\implies\angle DCA=\pi/6,\quad\angle DCB=\pi/6+\pi/3=\pi/2$
Use base area and perpendicular height.
$[BCD]=\frac12(2)(\sqrt3)=\sqrt3,\quad V=\frac13[BCD]\,PA=\frac{\sqrt3}3$

Think first. Reveal a hint when the class is ready.

2024 JM02