Shadow Problem Similar Triangles

Run the Shadow Problem Similar Triangles MicroSim Fullscreen Edit with the p5.js Editor

About This MicroSim

A 6-foot person walks away from a 15-foot street lamp. Where is the equation? There is no Pythagorean theorem here and no volume formula. The equation is hiding in the shape of the picture.

The light ray that skims the top of the person's head and lands at the shadow tip is the slanted side of two right triangles at once:

Triangle	Height	Base
Big (blue): lamp post to shadow tip	15 ft	\(x + s\)
Small (orange): person to shadow tip	6 ft	\(s\)

Both triangles have a right angle at the ground, and both contain the same angle at the shadow tip. Two matching angles force the third to match as well, so the triangles are similar: same shape, different size. Similar triangles have equal ratios of matching sides, which gives the proportion:

\[\frac{15}{x + s} = \frac{6}{s}\]

Cross-multiplying gives \(15s = 6x + 6s\), so \(s = \frac{2}{3}x\). That single equation is what you differentiate to solve the related rates problem.

The purple arcs are matching angle marks. A single arc marks the angle the two triangles share at the shadow tip. Double arcs mark the equal angles at the top of the lamp and the top of the person's head.

Delta Moment

"Two triangles, one light ray. The little triangle is just a shrunken copy of the big one, so their slopes have to agree. Same tilt, different size. I know a thing or two about tilt!"

How to Use

Click Big Triangle to see only the triangle from the lamp post to the shadow tip. Notice that its base is \(x + s\).
Click Small Triangle to see only the triangle formed by the person and the shadow. Its base is just \(s\).
Click Both to see them together with their matching angle marks.
Drag the Distance x slider to walk the person away from the lamp.
Watch the proportion panel. The two fractions always have the same value, even though that value changes as the person moves.

What to Observe

As \(x\) grows, both triangles get longer and flatter, but they always stay the same shape as each other.
The two numeric fractions in the panel are always equal. That equality is the similar-triangle proportion.
The shadow is always two-thirds of the distance from the lamp: \(s = \frac{2}{3}x\).
The shadow tip is at \(x + s\), so it moves faster than the person does.

Iframe Embed Code

You can add this MicroSim to any web page by adding this to your HTML:

<iframe src="https://dmccreary.github.io/calculus/sims/shadow-problem-similar-triangles/main.html"
        height="537px"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

Grades 11-12 (college placement Calculus)

Duration

10-15 minutes

Learning Objectives

After using this MicroSim, students will be able to:

Explain why the lamp triangle and the person triangle are similar (Bloom Level 2)
Identify the height and base of each triangle, including why the big base is \(x + s\)
Set up the proportion \(\frac{15}{x+s} = \frac{6}{s}\) from the diagram
Describe how the shadow length depends on the distance from the lamp

Prerequisites

Similar triangles and proportions from geometry
Solving a proportion by cross-multiplying
The idea of a related rates problem: quantities that change together over time

Activities

Find the triangles (3 min): With Both selected, students point out the three angles of each triangle and say which pairs match. Then they check their answer using the Big Triangle and Small Triangle buttons.
Name the bases (3 min): Ask "What is the base of the big triangle?" Many students answer \(x\). Click Big Triangle and read the blue dimension line to see that it is \(x + s\).
Test the proportion (4 min): Students move the slider to three positions and record \(x\), \(s\), and both fraction values. They confirm the fractions always agree and that \(s/x\) is always \(\frac{2}{3}\).
Connect to rates (5 min): Using \(s = \frac{2}{3}x\) and \(\frac{dx}{dt} = 4\) ft/sec, students find \(\frac{ds}{dt}\) and then the speed of the shadow tip, \(\frac{d}{dt}(x + s)\).

Discussion Questions

Which side do the two triangles share part of? Which angle do they share completely?
Why is \(\frac{15}{x} = \frac{6}{s}\) the wrong proportion?
The value of each fraction changes as the person walks. Why does that not break the equation?
Does the shadow tip speed depend on where the person is standing? How can you tell from \(s = \frac{2}{3}x\)?

Assessment

Students have met the objective when they can:

sketch a lamp-and-shadow scene and outline both triangles
mark the right angles and the shared angle
write the correct proportion with \(x + s\) as the base of the large triangle
explain in a sentence why the triangles must be similar