Geometry and Measurement
Geometry and Measurement tests transformations and their effect on congruence and similarity, the Pythagorean theorem and its converse, distance in the coordinate plane, and volume of cylinders, cones, and spheres.
Transformations
Translations, reflections, and rotations are rigid motions: they preserve side lengths and angle measures, so the image is congruent to the preimage. A dilation with center and scale factor k preserves angle measures and multiplies every distance from the center by k, producing a similar figure. For k > 1 the image enlarges; for 0 < k < 1 it reduces. Under a dilation, side lengths scale by k, areas scale by k², and volumes scale by k³.
Congruence and similarity
Two figures are congruent if one can be mapped to the other by a sequence of rigid motions. They are similar if a sequence of rigid motions and dilations maps one to the other. Corresponding angles are equal in similar figures; corresponding side lengths are proportional.
Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c, a² + b² = c². The hypotenuse is always the side opposite the right angle and is the longest side. The converse is also true: if a² + b² = c² for the three sides, the triangle is right. The distance between points (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²).
Volume
Cylinder: V = πr²h. Cone: V = (1/3)πr²h. Sphere: V = (4/3)πr³. Use the radius, not the diameter; if the diameter is given, halve it first.
Worked example
A right triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg. Use 5² + b² = 13², so 25 + b² = 169. Subtract: b² = 144. Since length is positive, b = 12 cm. Check: 5² + 12² = 25 + 144 = 169 = 13².
Traps that cost marks
- Using a leg as c in the Pythagorean theorem.
- Applying a² + b² = c² to a triangle that is not right.
- Scaling area by k instead of k², or volume by k instead of k³.
- Using diameter in a volume formula without halving it.
- Reversing coordinate signs when applying a reflection or rotation rule.
- Forgetting that a dilation does not preserve side length, only shape.