ShieldsGeometry
Spatial GeometryPlane GeometryCoordinates & VectorsApplied & ExamExam papers

Shields Geometry | Geometry lessons, formulas & exam practice

Clear geometry notes for Hong Kong students — from junior secondary foundations through HKDSE-style timed practice. Each lesson includes formulas, a worked example and short drills.

Featured lessons

All lessons

All lessons (100)

Page 2/2
51Basic proportionality (Thales) theoremPlane Geometry · Ms Ho · AD/DB = AE/EC52Angles in the same segmentPlane Geometry · Ms Ho · angles in the same segment are equal53Rhombus diagonals are perpendicular bisectorsPlane Geometry · Ms Ho · d₁² + d₂² = 4a²54Area of a trapeziumPlane Geometry · Ms Tam · area = ½(a+b)h55Sine rule for a non-right trianglePlane Geometry · Mr Wong · a/sin A = b/sin B = c/sin C56Cosine rule to find a sidePlane Geometry · Mr Wong · c² = a² + b² − 2ab cos C57Angle bisector theoremPlane Geometry · Ms Ho · BD/DC = AB/AC58Two tangents from an external pointPlane Geometry · Ms Ho · PT = QT59Distance between two pointsCoordinates & Vectors · Mr Wong · d = √[(x₂−x₁)²+(y₂−y₁)²]60Midpoint and section formula on a lineCoordinates & Vectors · Mr Wong · M = ((x₁+x₂)/2,(y₁+y₂)/2)61Equation of a line through two pointsCoordinates & Vectors · Mr Wong · m = (y₂−y₁)/(x₂−x₁)62Parallel and perpendicular gradientsCoordinates & Vectors · Mr Wong · parallel: m₁=m₂63Area of a triangle from coordinatesCoordinates & Vectors · Mr Wong · area = ½|det form|64Circle equation from centre and radiusCoordinates & Vectors · Mr Wong · (x−h)²+(y−k)²=r²65Intersection of a line and a circleCoordinates & Vectors · Mr Wong · discriminant >0 two points66Vector addition as a parallelogramCoordinates & Vectors · Mr Wong · a + b = diagonal67Position vectors and midpointsCoordinates & Vectors · Mr Wong · OM = (a+b)/268Rotation by 90° about the originCoordinates & Vectors · Ms Ho · 90° ACW: (x,y)→(−y,x)69Enlargement from a centreCoordinates & Vectors · Ms Ho · X′ = C + k(X − C)70Locus: perpendicular bisectorCoordinates & Vectors · Mr Wong · |PA|=|PB| ⇒ P on perp. bisector of AB71Proving collinearity with vectorsCoordinates & Vectors · Mr Wong · AB = k AC72Dot product and right anglesCoordinates & Vectors · Mr Wong · a · b = |a||b|cos θ73Parametric equation of a lineCoordinates & Vectors · Mr Wong · r = a + t d74Translation by a vectorCoordinates & Vectors · Ms Ho · (x,y) → (x+a,y+b)75Coordinate proof that a quadrilateral is a parallelogramCoordinates & Vectors · Mr Wong · midpoint AC = midpoint BD76Shortest path via a mirror lineCoordinates & Vectors · Ms Ho · reflect → straight line → intersect mirror77Gradient as tan of the inclinationCoordinates & Vectors · Mr Wong · m = tan θ78Dividing a segment in a vector ratioCoordinates & Vectors · Mr Wong · p = (n a + m b)/(m+n)79Command words: find, show, explain, henceApplied & Exam · Mr Wong · read the verb before calculating80Annotating the exam diagram in the first minuteApplied & Exam · Ms Tam · given → mark → plan → calculate81A 30-second unit and scale checkApplied & Exam · Mr Wong · length / area / volume → unit82Map scales and area factorsApplied & Exam · Ms Tam · area factor = (length factor)²83Back bearings and reverse journeysApplied & Exam · Mr Wong · back bearing = bearing ± 180° (mod 360°)84Percentage error in a measured lengthApplied & Exam · Ms Tam · % error = (|approx − true|/true)×100%85Two-pass timing for a mixed geometry paperApplied & Exam · Mr Wong · pass 1 easy marks → pass 2 constructions86Roof pitch as a right triangleApplied & Exam · Ms Tam · tan θ = rise/run87Ladder against a wallApplied & Exam · Ms Ho · ladder² = height² + base²88Heights from shadows with similar trianglesApplied & Exam · Ms Tam · h₁/s₁ = h₂/s₂89Water tank problems with consistent unitsApplied & Exam · Ms Tam · 1 m³ = 1000 L90Keeping a one-week geometry error logApplied & Exam · Mr Wong · classify → correct → retest91Choosing formulas by the quantity namedApplied & Exam · Mr Wong · target quantity → formula family92Exact trig values you should not calculatorApplied & Exam · Ms Ho · sin30=1/293Paint coverage from surface areaApplied & Exam · Ms Tam · litres = area / coverage94Layout that earns method marksApplied & Exam · Mr Wong · formula → substitute → simplify → conclude95Compound shapes: add and subtract areasApplied & Exam · Ms Ho · A = A₁ + A₂ − overlap96Spider-and-fly shortest path on wallsApplied & Exam · Mr Wong · shortest path = straight line on a suitable net97Rounding lengths without wrecking the next stepApplied & Exam · Mr Wong · round at the end98Calculator degree mode and inverse trigApplied & Exam · Ms Ho · θ = tan⁻¹(opp/adj) in DEG mode99Arc length and sector areaPlane Geometry · Ms Ho · arc = 2πr·(θ/360°)100Estimating before calculatingApplied & Exam · Ms Tam · estimate → calculate → compare