Track / Stage 4 / 141 of 340 #401 of 2000

Problem 401

AMC 12 late, AIME early

Geometry Difficulty 4.7 Find the answer

2. Given $P$ is the center of the upper base $\triangle A_{1} B_{1} C_{1}$ of a regular triangular prism $A B C-A_{1} B_{1} C_{1}$ , construct a plane $B C D \perp A P$ , intersecting the edge $A A_{1}$ at point $D$ . If $A A_{1}=2 A B=2$ , then the volume of the tetrahedron $D-A B C$ is ( ).
(A) $\frac{\sqrt{3}}{48}$
(B) $\frac{\sqrt{3}}{24}$
(C) $\frac{\sqrt{3}}{16}$
(D) $\frac{\sqrt{3}}{12}$

Your answer

Official solution

2. A.

As shown in Figure 3, let the plane $A A_{1} P$ intersect the edges $B C$ and $B_{1} C_{1}$ at points $E$ and $E_{1}$ , respectively.
In the rectangle $A E E_{1} A_{1}$ , $A A_{1}=2$ ,
$A E=A_{1} E_{1}=\frac{\sqrt{3}}{2}$ ,
$A_{1} P=\frac{\sqrt{3}}{3} \text {. }$

From $\frac{A A_{1}}{A E}=\frac{A P}{E D}$ , we get

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.