Geometry: Transformations, Similarity, Right Triangles, Circles, and Measurement
What this domain tests
Ohio's Geometry standards emphasize transformation-based reasoning, similarity, right-triangle trigonometry, circle relationships, coordinate geometry, and measurement. OST items often ask you to justify why figures are congruent or similar, apply a theorem to find a length or angle, or interpret a model. Final answers matter, but so does choosing the correct relationship.
Core rules
- Rigid motions-translations, rotations, reflections-preserve distance and angle measure, so they establish congruence. A dilation with scale factor k multiplies lengths by k but preserves angle; dilation plus rigid motion establishes similarity.
- Triangle similarity: AA, SSS, and SAS similarity. Corresponding side ratios are equal; areas scale by k², volumes by k³.
- Right triangles: Pythagorean theorem a²+b²=c². In 45-45-90, sides are x, x, x√2. In 30-60-90, sides are x, x√3, 2x.
- Trigonometric ratios: sin=opposite/hypotenuse, cos=adjacent/hypotenuse, tan=opposite/adjacent. Inverse trig finds an angle.
- Circle: an inscribed angle is half its intercepted central angle. Arc length = (θ/360)2πr; sector area = (θ/360)πr². A tangent is perpendicular to the radius at the point of tangency. Circle equation: (x−h)²+(y−k)²=r².
- Coordinate geometry: parallel lines have equal slopes; perpendicular slopes are negative reciprocals. The midpoint formula averages coordinates; the distance formula follows from the Pythagorean theorem.
- Volume: cylinder V=πr²h, cone V=⅓πr²h, sphere V=4/3πr³. Cross-sections parallel to a base match the base; density compares mass to volume.
Worked example
A circle has equation (x−3)²+(y+2)²=49. Find its center and radius, then the length of a 90° arc. Rewrite y+2 as y−(−2), so center (3,−2); r²=49, r=7. Arc length = (90/360)(2π·7)=¼·14π=3.5π ≈ 11.0.
Traps
Do not assume diagrams are to scale. In similarity, match corresponding vertices, not just the longest sides. For arc length use circumference, not area. Check whether an angle is inscribed or central before halving. Negative signs in circle equations flip the center's coordinates. If a scale factor is given for a solid, apply k² to area or k³ to volume, not k.