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Similarity & Triangle Trigonometry Lab

Explore triangle similarity, right triangle trigonometry with SOH CAH TOA, and oblique triangle solving with the Law of Sines and Law of Cosines. Detect the ambiguous case, collect data, and build lab reports.

Guided Experiment: SOH CAH TOA Investigation

If you change the size of a right triangle but keep the same acute angle, what happens to the sine, cosine, and tangent ratios?

Write your hypothesis in the Lab Report panel, then click Next.

Triangle Diagram

Triangle 1 (ABC)Triangle 2 (DEF)Similar (k = 1.000)

Controls

Drag the triangle vertices on the diagram to test similarity. The system checks AA, SAS, and SSS criteria automatically.

Results

Triangle 1 (ABC)

Sides: 3.61, 3.61, 4.00

Angles: 56.3°, 56.3°, 67.4°

Area: 6.00

Triangle 2 (DEF)

Sides: 3.61, 3.61, 4.00

Angles: 56.3°, 56.3°, 67.4°

Area: 6.00

Similarity Tests

AA
SAS
SSS
Scale factor k=1.000k = 1.000
Side ratios: 1.000 : 1.000 : 1.000

Data Table

(0 rows)
#TrialModeSide aSide bSide cAngle A(°)Angle B(°)Angle C(°)Area
0 / 500
0 / 500
0 / 500

Reference Guide

Triangle Similarity

Two triangles are similar if they have the same shape (but not necessarily the same size). Three tests prove similarity.

  • AA (Angle-Angle) Two pairs of equal angles
  • SAS (Side-Angle-Side) Two pairs of proportional sides with equal included angle
  • SSS (Side-Side-Side) All three pairs of sides in the same ratio
△ABC∼△DEF  ⟹  ABDE=BCEF=ACDF=k\triangle ABC \sim \triangle DEF \implies \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

SOH CAH TOA

The three primary trigonometric ratios relate the sides of a right triangle to its acute angles.

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Law of Sines

Relates each side of a triangle to the sine of its opposite angle. Useful for AAS, ASA, and SSA cases.

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Ambiguous case (SSA) When you know two sides and an angle opposite one of them, zero, one, or two triangles may exist.

Law of Cosines

A generalization of the Pythagorean theorem for any triangle. Useful for SAS and SSS cases.

c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab\cos(C)

When C = 90°, this reduces to the Pythagorean theorem since cos(90°) = 0.