Geometry and Measurement
The Geometry and Measurement domain covers Euclidean reasoning, coordinate geometry, transformations, and measurement of two- and three-dimensional figures.
Triangles and circles
Triangles are congruent by SSS, SAS, ASA, and AAS; SSA is not a congruence criterion. Similarity requires AA, SSS~, or SAS~. The Pythagorean theorem and its converse link side lengths to right angles. In a circle, an inscribed angle is half its intercepted arc, a tangent is perpendicular to the radius at the point of tangency, and sector area is (θ/2)r² when θ is measured in radians.
Coordinates and transformations
Distance is √((x₂−x₁)² + (y₂−y₁)²) and the midpoint averages the coordinates. Rigid motions preserve distance and angle measure; dilations preserve shape and scale lengths by k.
Scaling and measurement
Under a similarity ratio k, lengths scale by k, areas by k², and volumes by k³. This is the most frequently tested measurement relationship.
Worked example
A right cone has radius 3 cm and height 4 cm. The slant height is √(3² + 4²) = 5. Lateral area is πrl = 15π and base area is 9π, so total surface area is 24π cm². Volume is (1/3)πr²h = (1/3)π(9)(4) = 12π cm³. Doubling every dimension multiplies area by 4 and volume by 8.
Traps
Applying the similarity ratio directly to area or volume; mixing arc length with sector area; using a cone's height where its slant height is required; treating SSA as a congruence shortcut.