Japan 高校1年 Mathematics A
Chapters: 3
1. Properties of figures
Triangle properties (centres, Ceva, Menelaus) · Circle properties (tangents, cyclic quadrilaterals) · Solid figures (Euler's formula)
- Properties of Triangles and Their Centres – The angles of every triangle add up to 180°. A triangle has four famous centres. Medians meet at the centroid G, which cuts each median 2 : 1. Perpendicular bisectors meet at the circumcentre O, the centre of the circle through the corners. Angle bisectors meet at the incentre I, the centre of the circle inside that touches all sides. Altitudes meet at the orthocentre H. O, G and H lie on one line, the Euler line.
- Tangent to a Circle – A tangent is a line that touches a circle at exactly one point. At that point it makes a 90° angle with the radius, and two tangents drawn from one outside point are always equal in length.
2. Counting and probability
Counting elements of sets · Permutations and combinations · Probability laws · Independent trials · Conditional probability
- Sets: Representation, Types, Subsets, Venn Diagrams and Operations – A set is a well-defined collection of different objects. We write it in roster form {2, 4, 6} or set-builder form {x : x is even}. Sets can be empty, finite, infinite or equal. If every element of B is in A, B is a subset of A (B ⊂ A); a set with n elements has 2ⁿ subsets. Intervals like (a, b) and [a, b] are subsets of real numbers. The universal set U holds everything under study. With Venn diagrams we see union A ∪ B, intersection A ∩ B, difference A − B and complement A′ = U − A.
- 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.
- 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).
3. Mathematics and human activity
Number and figure concepts in human culture (divisors, base-n, Euclidean algorithm) · Math history, puzzles and games
- Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.