Shadow of a Pole

Geometry Level 3

A vertical pole extends from the origin $(0, 0, 0)$ to the point $(0, 0, 1)$ . A point light source is located at the point $(1, 2, 4)$ .

What is the length of the shadow of the pole in the $xy$ -plane ?

1 + \sqrt {2}

2 \sqrt {2} - 1

5 / \sqrt{2}

\sqrt {5} / 3

1 solution

Unstable Chickoy
Jun 9, 2014

Bear with the notations.

$O$ (0, 0, 0) - origin.

$P$ (0, 0, 1) - highest point of the pole.

$L$ (1, 2, 4) - light source.

$M$ - a point directly below the light source $L$ perpendicular to the point $P$ .

$S$ - point from the tip of the shadow, perpendicular to the origin, lies in the $XY$ plane.

$\triangle{LPM}$ is similar to $\triangle{OSP}$

$\frac{\overline{OS}}{\overline{OP}} = \frac{\overline{MP}}{\overline{ML}}$

$\frac{\overline{OS}}{1} = \frac{\sqrt{1^2 + 2^2}}{4 - 1}$

$\overline{OS} = \boxed{\frac{\sqrt{5}}{3}}$

Good use of similar triangles. Even though we were given 3-D coordinates, it ended up just being a 2-D problem.

Being able to reduce the dimension of the problem makes it much easier to approach.

Calvin Lin Staff - 4 years, 7 months ago

Thank you. I already forgot this website quite a while ago :) . I wish I could get back my spare time, same as the old days

Unstable Chickoy - 4 years, 5 months ago

You should :)

Brilliant's community works by having numerous people contribute wherever they can. A problem here, a solution there, it all adds up!

Calvin Lin Staff - 4 years, 5 months ago