Similarity, Right Triangles, and Trigonometry
Similarity, Right Triangles, and Trigonometry
This domain tests proportional reasoning with similar figures, the Pythagorean theorem, special right triangles, and right-triangle trigonometry, including applications with angles of elevation and depression.
Core rules
- Triangles are similar by AA, SSS similarity, or SAS similarity. Corresponding angles are congruent and corresponding sides are proportional.
- If the scale factor from figure A to figure B is k, then lengths scale by k, areas by k squared, and volumes by k cubed.
- Pythagorean theorem: a squared plus b squared equals c squared, where c is the hypotenuse. Converse: if a squared plus b squared equals c squared, the triangle is right.
- Special right triangles: 45-45-90 has sides x, x, and x root 2; 30-60-90 has sides x, x root 3, and 2x, with x opposite the 30-degree angle.
- Trig ratios: sin theta equals opposite over hypotenuse, cos theta equals adjacent over hypotenuse, tan theta equals opposite over adjacent. The altitude to the hypotenuse creates geometric-mean relationships.
Worked example
A surveyor stands 40 meters from the base of a vertical building. The angle of elevation to the top is 32 degrees. Find the height. The building height is opposite the 32-degree angle, and 40 m is the adjacent side. tan 32 degrees equals h divided by 40, so h equals 40 tan 32 degrees, which is about 40 times 0.6249, or 25.0 m. The angle of elevation is measured from horizontal, so the horizontal distance is adjacent, not the line of sight.
Traps
Set your calculator to degree mode. Choose opposite and adjacent relative to the angle you know, not the triangle's orientation. Do not use the Pythagorean theorem on non-right triangles. In similarity, do not apply the scale factor once to area; area changes by k squared. In a 30-60-90 triangle, the longer leg is x root 3, not x root 2. Keep the line of sight separate from horizontal distance in elevation problems.