Geometry
The Geometry domain covers congruence and similarity, right-triangle trigonometry, circles, coordinate geometry, and volume. MCAS Grade 10 expects you to justify relationships, not just compute, so connect each formula to a diagram or transformation.
Core rules
- Rigid motions-translations, rotations, reflections-preserve distance and angle, so they establish congruence. Dilations create similar figures.
- Triangle congruence: SSS, SAS, ASA, AAS, and HL. SSA and AAA do not guarantee congruence.
- Similar figures: corresponding sides are proportional. If the scale factor is k, areas scale by k² and volumes by k³.
- Right triangles: a² + b² = c²; sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x.
- Circles: a central angle equals its intercepted arc; an inscribed angle is half its intercepted arc. The circle (x − h)² + (y − k)² = r² has center (h, k) and radius r.
Worked example
A circle has radius 6 and a central angle of 120°. Find the length of the intercepted arc and the measure of an inscribed angle that subtends the same arc. Arc length is (120/360)·2π(6) = (1/3)·12π = 4π. An inscribed angle is half the central angle, so it measures 60°. The arc measure is 120°, but arc length is a distance, so it carries units, not degrees.
Traps
- Do not use SSA as a congruence shortcut; the ambiguous case can produce two triangles.
- Area scale factor is k², not k; volume scale factor is k³.
- Check whether a length is a radius or a diameter before applying a formula.
- Match trig ratios to the reference angle; opposite and adjacent swap when the angle changes.