In pyramid P–ABC, PC⊥plane ABC, PC=3, ∠BCA=π/2. D lies on AB and E on BC; CD=DE=√2, CE=2 and EB=1. Find AC.

01 / Standard#Your turn

Official paper · jm02-2022 · 1(a) · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Let M be the midpoint of CE.

Hint 2

Compare the similar right triangles DMB and ACB.

Worked solution

The isosceles triangle CDE gives DM⊥CE and DM=1.
$CM=ME=1,\quad DM=\sqrt{DE^2-ME^2}=\sqrt{2-1}=1$
MB=2 and CB=3; the similar triangles have the same angle at B.
$\frac{AC}{CB}=\frac{DM}{MB}=\frac12\implies AC=3/2$

AC=3/2.

Checks and common pitfalls: MB includes ME and EB, so MB=2.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The isosceles triangle CDE gives DM⊥CE and DM=1.
$CM=ME=1,\quad DM=\sqrt{DE^2-ME^2}=\sqrt{2-1}=1$
MB=2 and CB=3; the similar triangles have the same angle at B.
$\frac{AC}{CB}=\frac{DM}{MB}=\frac12\implies AC=3/2$

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

2022 JM02