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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

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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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Keep Going

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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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