Investigating Proportionality — Visual Lab

Module 3 asks one big question: when you shrink or stretch a right triangle, what stays the same? Drag the triangle's angle below and watch two similar triangles — one big, one small — report the exact same sine, cosine, and tangent. That constancy is the whole reason right-triangle trigonometry works.

Module 03 · Similarity & Trig · TEKS G.7 / G.8

📖 Student Edition for this module (PDF)

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Similarity & Right-Triangle Trigonometry

Live tool — drag the dot, slide the angle, change the scale factor. The math updates in real time.

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What You're Seeing & What to Try

A two-minute orientation, then four investigations to run yourself.

What you're seeing

A right triangle with the reference angle \(\theta\) at the bottom-left and the right angle at the bottom-right.
A second, similar triangle sharing that same angle \(\theta\), scaled by a factor \(k\) you control — same shape, different size.
The three trig ratios computed live: \(\sin\theta=\tfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\tfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\tfrac{\text{opp}}{\text{adj}}\).
Side lengths for each triangle, so you can see the lengths differ while the ratios match.

Try this

Set \(\theta=30^\circ\) with the preset. Check that \(\sin 30^\circ = 0.5\) exactly.
Drag the scale factor \(k\) from \(1.2\) to \(2.4\). Watch every side length change — but the three ratios never move.
Drag the dot at the top of the triangle. As \(\theta\) grows, which ratio climbs fastest? (Hint: watch \(\tan\theta\).)
Find the angle where \(\sin\theta \approx \cos\theta\). Why does that happen near \(45^\circ\)?

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Key Vocabulary & Standards

The language behind the lab, and the TEKS it builds.

Similar figures: Same shape, possibly different size. Corresponding angles are equal and corresponding sides are proportional.
Scale factor (k): The constant ratio between corresponding sides of two similar figures. Here, triangle 2 = \(k\times\) triangle 1.
AA similarity: If two angles of one triangle equal two angles of another, the triangles are similar. Both triangles here share \(\theta\) and a \(90^\circ\) angle.
Trigonometric ratio: For a fixed acute angle in a right triangle, the ratios opp/hyp, adj/hyp, and opp/adj are constant — that is sine, cosine, and tangent.

G.7A G.7B G.8A G.8B

G.7 — apply similarity and AA criteria · G.8 — apply the trigonometric ratios sine, cosine, and tangent to solve right triangles.

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Course Syllabus

Policies, grading, the studio learning environment, and the full itinerary by grading period.

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Pacing Guide

Every module mapped across the grading periods — see where Module 3 falls in the year.

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