Geometry: Transformations, Right Triangles, and Circles
What this domain tests
Geometry on this exam mixes rigid-motion reasoning with measurement. Expect transformations and the definition of congruence and similarity, triangle theorems and proof, right-triangle trigonometry, coordinate geometry (distance, midpoint, slope, parallel and perpendicular lines), circle relationships, and area and volume including composite figures. Because Georgia items pair a figure with a reason, expect to justify a step, not just compute a number.
Core rules
- Rigid motions preserve distance and angle: translations, rotations, and reflections produce congruent figures. Dilations preserve angle and shape but scale length by a factor k, so areas scale by k² and volumes by k³.
- Similar triangles have equal corresponding angles and proportional sides. AA is sufficient; SSS and SAS similarity require proportional sides.
- Right triangles: a² + b² = c² with c the hypotenuse. Trigonometry: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Choose a ratio by naming the sides relative to the angle in question.
- Circle facts: an inscribed angle is half its intercepted arc; a central angle equals its arc; a tangent is perpendicular to the radius at the point of tangency; the equation (x − h)² + (y − k)² = r² encodes center (h, k) and radius r.
- Arc length and sector area are fractions of the whole: arc = (θ/360)(2πr) and sector area = (θ/360)(πr²) for θ in degrees.
- Coordinate tools: distance = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint averages the coordinates; parallel lines share a slope; perpendicular slopes are negative reciprocals.
Worked example
A circle has radius 9 and a sector with central angle 80°. Find the arc length and the sector area.
The sector is 80/360 = 2/9 of the circle. Circumference is 2π(9) = 18π, so arc length = (2/9)(18π) = 4π ≈ 12.57. Area of the whole circle is π(9²) = 81π, so sector area = (2/9)(81π) = 18π ≈ 56.55. Notice the two answers use different formulas; substituting the area formula into an arc-length question is a frequent and costly slip.
Traps that cost marks
- Using diameter where radius is required, or the reverse.
- Computing arc length with πr² or sector area with 2πr.
- Selecting sin and cos from memory instead of from the sides relative to the given angle.
- Scaling area by k rather than k² when a figure is dilated.
- Confusing the angle of elevation with the angle of depression, which sit on opposite horizontal lines.