Topic A: Coordinate Geometry
Lessons 1–5 · Distance, slope, and proving properties with coordinates.
01–03 — Lessons 1–3
Slope & Parallel/Perpendicular Lines
- Slope formula: m = (y₂−y₁)/(x₂−x₁).
- Parallel lines: equal slopes (m₁ = m₂).
- Perpendicular lines: slopes are negative reciprocals (m₁ · m₂ = −1).
- Horizontal line: slope = 0. Vertical line: undefined slope.
- Use coordinates to prove that a quadrilateral is a special type.
Slope of AB = 2/3. A perpendicular line has slope −3/2.
Practice — Lessons 1–3
Q1 · Multiple Choice
Line 1 has slope 4. A perpendicular line has slope:
Q2 · Short Answer
Are the lines through (0,0)→(3,6) and (0,0)→(6,3) perpendicular? (slopes: 2 and 1/2)
04–05 — Lessons 4–5
Coordinate Proofs
- Place figures strategically on the coordinate plane to simplify proofs.
- Common placements: put one vertex at origin, one side along an axis.
- Use distance formula to show sides are congruent.
- Use slope formula to show sides are parallel or perpendicular.
- Use midpoint formula to show diagonals bisect each other.
To prove a quadrilateral is a parallelogram: show both pairs of opposite sides have equal slopes.
Practice — Lessons 4–5
Q3 · Multiple Choice
To prove a quadrilateral is a rectangle using coordinates, you show:
Q4 · Multiple Choice
Placing a right angle at the origin in a coordinate proof is helpful because:
Topic B: Area & Perimeter of Polygons
Lessons 6–10 · Calculating and comparing polygon measurements.
06–08 — Lessons 6–8
Area Formulas
- Triangle: A = (1/2)bh.
- Parallelogram: A = bh.
- Trapezoid: A = (1/2)(b₁+b₂)h.
- Regular polygon: A = (1/2)·apothem·perimeter.
- On coordinate plane: use the Shoelace formula or decompose into triangles/rectangles.
- All altitudes must be perpendicular to the base.
Trapezoid with b₁=6, b₂=10, h=4: A = (1/2)(16)(4) = 32 sq units.
Practice — Lessons 6–8
Q5 · Short Answer
Find the area of a triangle with base 14 and height 9.
Q6 · Multiple Choice
A trapezoid has parallel bases 8 and 12, and height 5. Area = ?
09–10 — Lessons 9–10
Perimeter & Composite Figures
- Perimeter: sum of all side lengths.
- For composite figures: break into simpler shapes, find each area/perimeter.
- When a region is 'removed,' subtract its area from the whole.
- Coordinate perimeter: use the distance formula for each side.
- Regular polygon perimeter: n × side length.
L-shaped figure: find total rectangle area, subtract the missing corner area.
Practice — Lessons 9–10
Q7 · Multiple Choice
A composite figure is formed by adding a triangle to a rectangle. Its area is:
Q8 · Short Answer
A regular hexagon has side length 6. Its perimeter is:
Topic C: Interior & Exterior Angles
Lessons 11–15 · Angle sums in polygons.
11–15 — Lessons 11–15
Polygon Angle Sums
- Interior angle sum of n-gon: S = (n−2)·180°.
- Each interior angle of regular n-gon: (n−2)·180° / n.
- Exterior angle sum: always 360° for any convex polygon.
- Each exterior angle of regular n-gon: 360°/n.
- Interior + exterior angle at each vertex = 180°.
Pentagon (n=5): interior sum = (3)(180°) = 540°. Each angle in regular pentagon = 108°.
Practice — Lessons 11–15
Q9 · Short Answer
What is the interior angle sum of an octagon (8 sides)?
Q10 · Multiple Choice
Each exterior angle of a regular hexagon measures: