Study guide: Geometry: Similarity and Right-Triangle Ratios
Similarity preserves angle measures and corresponding length ratios. Right triangles with the same acute angle are similar, explaining why sine, cosine and tangent depend on an angle rather than the triangle’s overall size.
Ideas to explain in your own words
- Corresponding side lengths of similar triangles share one scale factor. Find that factor from a known corresponding pair, then apply it to the requested matched side. Correspondence must be established before forming a ratio.
- For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length. Multiplying the hypotenuse by the sine recovers the opposite leg. The opposite side changes when the reference angle changes.
- Cosine compares the leg touching an acute reference angle with the hypotenuse. A supplied cosine ratio and hypotenuse determine that adjacent leg by multiplication. The hypotenuse also touches the acute angle but is not the adjacent leg in this definition.
- Tangent is the ratio of the opposite leg to the adjacent leg for an acute angle. It links the two legs directly, so a hypotenuse measurement is not needed. Keep the reference angle fixed while identifying the legs.
Useful vocabulary
- similarity
- Equal corresponding angles and proportional corresponding lengths.
- sine
- Opposite leg divided by hypotenuse in a right triangle.
- cosine
- Adjacent leg divided by hypotenuse in a right triangle.
- tangent
- Opposite leg divided by adjacent leg in a right triangle.
Watch your reasoning
- Correspondence must be established before forming a ratio. Keep larger-to-smaller or smaller-to-larger order consistent across pairs; mixing those orders creates an incorrect reciprocal scale factor.
- The opposite side changes when the reference angle changes. The hypotenuse remains the side opposite the right angle; do not use an adjacent leg in the sine denominator.
- The hypotenuse also touches the acute angle but is not the adjacent leg in this definition. Identify the two legs first and keep the right-triangle condition explicit.
A worked review example
Complete supplied guide examples: Use a Matched Length Ratio — Two similar triangles have corresponding sides 6 and 12. A second side on the smaller triangle is 8. Find its corresponding larger side. Candidate answers: (A) 15 cm (B) 16 cm (C) 17 cm (D) 32 cm. The larger-to-smaller length ratio is 12/6=2. Multiply the corresponding smaller side 8 by 2 to obtain 16. Match corresponding sides before writing the ratio. Use Opposite over Hypotenuse — In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 4 cm. Find its opposite side. Candidate answers: (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm. Sine is opposite/hypotenuse. If o/4=1/2, then o=4/2=2 cm. The side opposite θ depends on which acute angle was chosen. Use Adjacent Leg over Hypotenuse — For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 10 cm. Find the adjacent leg. Candidate answers: (A) 16 cm (B) 7 cm (C) 8 cm (D) 9 cm. Cosine is adjacent leg/hypotenuse. Multiply 10 by 4/5 to get 8. The hypotenuse is not counted as the adjacent leg in this ratio. Use Opposite over Adjacent Leg — For an acute angle θ, tan θ=3/4 and its adjacent leg is 8 cm. Find its opposite leg. Candidate answers: (A) 12 cm (B) 5 cm (C) 6 cm (D) 7 cm. Tangent is opposite/adjacent. Solve o/8=3/4, giving o=8×3/4=6 cm. No hypotenuse is used in this ratio.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.