United Grade 10 Geometry
Chapters: 6
1. Congruence, proof and constructions
Transformations in the plane · Congruence through rigid motions · Proving geometric theorems · Geometric constructions
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
- Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.
- Logic and Proof: How Mathematicians Show Something Is Always True – A proof is a chain of correct reasons that shows a statement is true in every case. Statements are true or false; 'if P then Q' (P ⇒ Q) means Q must hold whenever P does, and its converse Q ⇒ P may not. P ⇔ Q means both directions hold. Main methods: direct proof (from known facts step by step, often with algebra like 2n + 1 for odd numbers); proof by exhaustion (check every possible case); disproof by counterexample (one failing case kills a claim); proof by contradiction (assume the opposite and reach something impossible). Geometric proofs use given facts, definitions and theorems (parallel lines, congruent and similar triangles), with a reason for every line. Analytic proofs use coordinates; synthetic proofs use pure geometry.
- Geometric Constructions: Drawing Exactly with Compass and Straightedge – A geometric construction draws a figure exactly using only a compass and a straightedge (a ruler used for straight lines). Key constructions: the perpendicular bisector of a segment, the bisector of an angle, a perpendicular from a point to a line, angles of 60°, 30°, 90° and 45°, a triangle from three sides (SSS), two sides and the included angle (SAS) or two angles and a side (ASA), and regular polygons such as the hexagon. Each works because equal compass arcs make equal lengths, which give congruent triangles.
2. Similarity, proof and trigonometry
Similarity transformations · Proving similarity theorems · Right-triangle trigonometry · Trigonometry of general triangles
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
- Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
3. Extending to three dimensions
Volume formulas · 2D and 3D relationships · Modeling with geometry
- Surface Areas and Volumes: Combined Solids – Many real objects are two simple solids stuck together, like a cone on a cylinder. The surface area is only the outside skin you can touch, so the hidden joint is left out. The volume is the space inside, so you simply add the volumes of the parts.
- Cross-Sections and Solids of Rotation: Linking 2D and 3D Shapes – A cross-section is the flat face you get when a plane cuts a solid. A cube can give a square, rectangle, triangle, pentagon or hexagon; a cylinder, cone or sphere gives a circle when cut across. Spinning a 2D shape around an axis makes a solid of rotation: a rectangle makes a cylinder, a right triangle a cone, a semicircle a sphere. Cross-sections also explain volume: if two solids have equal cross-sections at every height, they have equal volume (Cavalieri's principle).
- Geometric Modelling: Describing Real Objects with Simple Solids – Geometric modelling means replacing a real object with simple shapes (cuboid, cylinder, cone, sphere, prism) so we can calculate with it. The cycle is: look at the real object, simplify it, measure, calculate volume or surface area, then check the answer against reality and improve the model. Scale changes lengths by k, areas by k² and volumes by k³. Geometry also helps us see patterns in nature and art, like hexagons in honeycombs.
4. Connecting algebra and geometry through coordinates
Coordinate geometry and proofs · Parabola from focus and directrix
- Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.
- Conic Sections (Class 11): Circle, Parabola, Ellipse and Hyperbola – Cutting a double cone with a plane gives a circle, an ellipse, a parabola or a hyperbola; a cut through the vertex gives a point, a line or a pair of lines (degenerate conics). Circle: (x − h)² + (y − k)² = r². Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1. Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², foci (±c, 0), e = c/a < 1, latus rectum 2b²/a, PF₁ + PF₂ = 2a. Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a, |PF₁ − PF₂| = 2a.
5. Circles with and without coordinates
Circle theorems · Arc length and sector area · Equation of a circle
- Circles: Chords and Angles – A chord joins two points on a circle. Longer chords make bigger angles at the centre, and equal chords make equal angles. The perpendicular from the centre to a chord cuts it in half, and equal chords are the same distance from the centre. The angle an arc makes at the centre is double the angle it makes anywhere on the rest of the circle, so the angle in a semicircle is 90°. In a cyclic 4-gon, opposite angles add to 180°.
- Area of Sector and Segment of a Circle – A sector is a pizza slice of a circle: its area is θ/360 × πr². Its crust is an arc of length θ/360 × 2πr. A segment is the slice minus the triangle inside it: segment = sector − triangle.
- Conic Sections (Class 11): Circle, Parabola, Ellipse and Hyperbola – Cutting a double cone with a plane gives a circle, an ellipse, a parabola or a hyperbola; a cut through the vertex gives a point, a line or a pair of lines (degenerate conics). Circle: (x − h)² + (y − k)² = r². Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1. Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², foci (±c, 0), e = c/a < 1, latus rectum 2b²/a, PF₁ + PF₂ = 2a. Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a, |PF₁ − PF₂| = 2a.
6. Applications of probability
Independence and conditional probability · Probability rules for compound events · Using probability for decisions · Permutations and combinations
- Probability: Events, Algebra of Events and Axioms – An event is any subset of the sample space S. From events A and B we build new events: not A (A′), A and B (A ∩ B), A or B (A ∪ B). Events are mutually exclusive if they share no outcome and exhaustive if together they cover S. The axiomatic approach says every P(E) ≥ 0, P(S) = 1, and for mutually exclusive A, B, P(A ∪ B) = P(A) + P(B). From these follow P(A′) = 1 − P(A) and P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.