Topic A: Pythagorean Theorem
Lessons 1–6 · Proof, Pythagorean triples, and distance in coordinates.
01–03 — Lessons 1–3
The Pythagorean Theorem
- Theorem: in a right triangle, a² + b² = c² (c = hypotenuse).
- Converse: if a² + b² = c², the triangle is a right triangle.
- Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and their multiples.
- Acute triangle: a² + b² > c². Obtuse: a² + b² < c².
- Used to find missing sides in right triangles.
a=5, b=12: c = √(25+144) = √169 = 13. It's a 5-12-13 triple.
Practice — Lessons 1–3
Q1 · Short Answer
Find the hypotenuse of a right triangle with legs 9 and 12.
Q2 · Multiple Choice
Triangle sides are 7, 10, and 12. Is it acute, right, or obtuse?
04–06 — Lessons 4–6
Distance Formula & Coordinate Geometry
- Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] — derived from Pythagorean theorem.
- Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2).
- Use to prove figures are squares, rectangles, parallelograms, etc., on a coordinate plane.
- Show perpendicular by: slopes are negative reciprocals (m₁ · m₂ = −1).
- Show parallel by: slopes are equal.
Distance from (1, 2) to (4, 6): d = √[(3)²+(4)²] = √25 = 5.
Practice — Lessons 4–6
Q3 · Short Answer
Find the distance from (0, 0) to (5, 12).
Q4 · Multiple Choice
The midpoint of (2, 8) and (6, 4) is:
Topic B: Special Right Triangles
Lessons 7–12 · 45-45-90 and 30-60-90 triangles.
07–09 — Lessons 7–9
45-45-90 Triangles
- 45-45-90 triangle: an isosceles right triangle.
- Leg : Leg : Hypotenuse = x : x : x√2
- Hypotenuse = leg × √2. Leg = hypotenuse / √2.
- Formed by cutting a square diagonally.
- Memorize the ratio — it speeds up many geometry problems.
Leg = 5: hypotenuse = 5√2. · Hypotenuse = 8: leg = 8/√2 = 4√2.
Practice — Lessons 7–9
Q5 · Short Answer
A 45-45-90 triangle has legs of length 6. What is the hypotenuse?
Q6 · Multiple Choice
In a 45-45-90 triangle, the hypotenuse is 10. What is each leg?
10–12 — Lessons 10–12
30-60-90 Triangles
- 30-60-90 triangle: Short leg : Long leg : Hypotenuse = x : x√3 : 2x
- Short leg (opp 30°) = hypotenuse / 2.
- Long leg (opp 60°) = short leg × √3.
- Hypotenuse = 2 × short leg.
- Formed by cutting an equilateral triangle in half.
Short leg = 4: long leg = 4√3, hypotenuse = 8. · Hypotenuse = 12: short leg = 6, long leg = 6√3.
Practice — Lessons 10–12
Q7 · Short Answer
In a 30-60-90 triangle, the short leg is 7. What is the hypotenuse?
Q8 · Multiple Choice
In a 30-60-90 triangle, which side is the longest?
Topic C: Trigonometric Ratios
Lessons 13–19 · SOH-CAH-TOA, inverse trig, and applications.
13–16 — Lessons 13–16
SOH-CAH-TOA
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
- Memory aid: SOH-CAH-TOA.
- Use a calculator in degree mode to find decimal values.
- Pythagorean identity: sin²θ + cos²θ = 1.
In a right △ with θ = 30°, hyp = 10: opp = 10·sin30° = 5, adj = 10·cos30° = 5√3.
Practice — Lessons 13–16
Q9 · Multiple Choice
In a right triangle, sin(θ) = 7/25. What is cos(θ)?
Q10 · Short Answer
A ladder leans against a wall, making a 60° angle with the ground. The ladder is 20 ft. How high up the wall? (sin60°=√3/2)
17–19 — Lessons 17–19
Inverse Trig & Angle of Elevation/Depression
- Inverse trig: sin⁻¹(x), cos⁻¹(x), tan⁻¹(x) — find the angle given a ratio.
- Angle of elevation: measured upward from horizontal.
- Angle of depression: measured downward from horizontal.
- Elevation and depression angles are alternate interior angles — they are equal.
- Draw a diagram, label, then write the trig ratio.
tan(θ) = 40/30 → θ = tan⁻¹(4/3) ≈ 53.1°.
Practice — Lessons 17–19
Q11 · Multiple Choice
You're standing 50 m from a tower. The angle of elevation to the top is 35°. Which equation finds the height h?
Q12 · Multiple Choice
The angle of depression from a cliff to a boat is 28°. The angle of elevation from the boat to the cliff is: