Geometry: Transformations, Angle Relationships, Pythagorean Theorem, and Volume
Grade 8 geometry covers transformations and their effect on congruence and similarity, angle relationships formed by parallel lines and transversals, the Pythagorean theorem and its converse, and the volume of cylinders, cones, and spheres. Items often ask you to justify why two figures are congruent or similar, or to find a missing length or angle.
Transformations
Translations, reflections, and rotations preserve side lengths and angle measures, so the image is congruent to the preimage. A dilation with scale factor k multiplies every length by k and preserves angles, producing a similar figure. Describing a sequence of rigid motions, plus a dilation when similarity is required, is a standard task.
Angle relationships
When two parallel lines are cut by a transversal, alternate interior angles are equal, corresponding angles are equal, and same-side interior angles are supplementary. The angles of any triangle sum to 180°, and an exterior angle equals the sum of the two remote interior angles.
Pythagorean theorem
For a right triangle with legs a and b and hypotenuse c: a² + b² = c². The converse also holds: if the sides satisfy a² + b² = c², the triangle is right. Common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25, and any multiple of a triple also works.
The distance between (x₁, y₁) and (x₂, y₂) follows from the theorem: d = √((x₂ − x₁)² + (y₂ − y₁)²).
Volume
- Cylinder: V = πr²h
- Cone: V = (1/3)πr²h
- Sphere: V = (4/3)πr³
A cone holds exactly one-third of the cylinder with the same base and height.
Worked example
Find the distance between (−2, 1) and (4, 9). The horizontal change is 4 − (−2) = 6 and the vertical change is 9 − 1 = 8. Then d = √(6² + 8²) = √(36 + 64) = √100 = 10.
Traps
- Applying a² + b² = c² to a triangle that is not right; verify with the converse first.
- Stopping at c² = 169 instead of taking the square root.
- Substituting the diameter for r in volume formulas.
- Mixing up the cone and cylinder formulas by dropping the 1/3 factor.
- Claiming a dilation changes angle measures; it does not.
- Forgetting that a reflection reverses orientation even though it preserves congruence.