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IGCSEFM Proof Dr J Frost ([email protected]) Objectives: (from the specification) Last modified: 22nd February 2016 Overview From GCSE, you should remember that a βproofβ is a sequence of justified steps, sometimes used to prove a statement works in all possible cases. Algebraic Proofs Geometric Proofs βProve that the sum of three consecutive even numbers is a multiple of 6.β ππ + ππ + π + ππ + π = ππ + π = π(π + π) which is a multiple of 6. ? Recall that the key at the end is to factorise out the 6. Prove that π¦ = π₯ (We will need to recap some circle theorems) Algebraic Proof Two common types of question: Prove that the difference between the squares of two consecutive odd numbers is a multiple of 8. Let numbers be ππ β π and ππ + π. π π ππ + π β ππ β π = πππ + ππ + π β πππ β ππ + π ? = πππ + ππ + π β πππ + ππ β π = ππ which is divisible by 8. We could have also used 2π + 1 and 2π + 3. Prove that π₯ 2 β 4π₯ + 7 > 0 for all π₯. ππ β ππ + π = πβπ πβπ+π = πβπ π+π ? π Since π β π β₯ π, thus π β π π +π>π Bro Hint: We know that anything βsquaredβ is at least 0. Could we perhaps complete the square? Test Your Understanding 1 [Specimen2 Q12] π is an integer. Prove that π β 2 2 + π 8 β π is always a multiple of 4. ππ β ππ + π + ππ β ππ = ππ + π ? = π(π + π) 2 3 [June 2013 P2 Q12] Prove that 5π + 3 π β 1 + π(π + 2) is a multiple of 3 for all integer values of π. = πππ + ππ β ππ β π + ππ + ππ = πππ β π ? = π πππ β π [Jan 2013 P1 Q5] π is a positive integer. (a) Write down the next odd number after 2π β 1 ππ +? π (b) Prove that the product of two consecutive odd numbers is always one less than a multiple of 4. ππ β π ππ + π = πππ β π ? of 4. πππ is a multiple 4 Prove that for all values of π₯, π₯ 2 β 6π₯ + 10 > 0 ππ β ππ + ππ = π β π π + π π β π π β₯ π thus ?π β π π + π > π 5 [Set 4 P1 Q16] Prove that, for all values of π₯, 2π₯ 2 β 8π₯ + 9 > 0 π π π π β ππ + π π =π πβπ πβπ+ π ?π π =π πβπ + π =π πβπ π+π π β π π β₯ π therefore π π β π π + π > π Identities What values of π₯ make the following equality hold true? π₯ is 2 or -2? π₯2 = 4 π₯2 β π₯ = π₯ π₯ β 1 π₯ could be?anything! ! The identity π(π₯) β‘ π(π₯) means that π π₯ = π(π₯) for all values of π₯. e.g. π₯2 β π₯ β‘ π₯ π₯ β 1 So π₯ 2 β‘ 4 would be wrong as it is not true when say π₯ is 1. When you have a quadratic/cubic/etc, all the coefficients must match to guarantee both sides of the identity are equal for all π. [Set 4 P1 Q2] In this identity, β and π are integer constants. 4 βπ₯ β 1 β 3 π₯ + β = 5 π₯ + π Work out the values of β and π πππ β π β ππ β ππ = ππ + ππ Comparing π terms: ππ β ?π = π β π = π Comparing constant terms: βπ β ππ = ππ β π = βπ Test Your Understanding [Set 3 P1 Q2] 5 3π₯ β 2 β 3 π₯ β β β‘ 4(ππ₯ + 2) Work out the values of β and π. πππ β ππ β ππ + ππ β‘ πππ + π πππ β ππ + ππ = πππ + π Comparing π terms: ππ = ππ β ?π = π Comparing constant terms: βππ + ππ = π β π = π AQA Worksheet (Algebraic Proof) BONUS QUESTIONS: 1 Prove algebraically that the sum of two consecutive odd numbers is divisible by 4. ππ β π + ππ + π = ππ which is divisible by 4. 3 ? ? 2 [GCSE] I think of two consecutive integers. Prove that the difference of the squares of these integers is equal to the sum of the two integers. Two numbers are: π and π + π Difference of squares: π + π π β ππ = ππ + π Sum of numbers: π + π + π = ππ + π These are equal. ? Prove that the difference between two consecutive cubes is one more than a multiple of 6. π + π π β ππ = ππ + πππ + ππ + π β ππ = πππ + ππ + π = ππ π + π + π The product of two consecutive integers is even, thus ππ(π + π) is divisible by 6. 4 Prove that the product of four consecutive numbers is one less than a square number. π π+π π+π π+π = ππ + π ππ + ππ + π = ππ + πππ + ππππ + ππ + π ? = ππ + ππ + π π Geometric Proof A recap of general angle theorems and Circle Theorems: ! Alternate angles ? are equal. ! Corresponding angles are equal. ? as βFβ angles) (Sometimes known π π + π = 180° π ! Vertically opposite ? angles are equal. ! Cointerior angles ? sum to πππ°. RECAP :: Circle Theorems Angle between radius and tangent is 90°. ? Opposite angles of cyclic quadrilateral are equal. ? Angle in semicircle is 90° ? Angles in same segment are equal. Tangents from a point to a circle are equal in length. ? ? Angle at centre is twice angle at circumference. ? Alternate Segment Theorem. ? Form of a Geometric Proof Set 1 Paper 1 Q8 ! Write statements in the form: β π΄π΅πΆ = π£πππ’π (ππππ ππ) β ππΆπ΅ = π₯ (base angles of isosceles triangle are equal) ? β π΅ππΆ = 2π₯ (angle at centre is double ? angle at circumference) Angles in Ξππ΅πΆ add to 180° β΄ π₯ + π₯ + 2π₯ = 180 ? 4π₯ = 180 π₯ = 45 β π΅ππΆ = 2π₯ = 90° Test Your Understanding Triangle π΄π΅πΆ is isosceles with π΄πΆ = π΅πΆ. Triangle πΆπ·πΈ is isosceles with πΆπ· = πΆπΈ. π΄πΆπ· and π·πΈπΉ are straight lines. a) Prove that angle π·πΆπΈ = 2π₯ β πͺπ©π¨ = π (base angles of isosceles triangle are equal) ? in π«π¨π©πͺ add to 180) β π¨πͺπ© = πππ β ππ (angles β π«πͺπ¬ = ππ (angles on straight line add to 180) b) Prove that π·πΉ is perpendicular to π΄π΅. πππβππ β π«π¬πͺ = π = ππ β π (base angles of isosceles triangle are equal) ? β π«ππ¨ = πππ β ππ β π β π = ππ° β΄ π«π is perpendicular to π¨π©. Last Step What do you think we would be the last step in your proof in each of these cases? π· Prove that π΄π΅πΆ is a straight line. πΆ β¦ β π¨π©π« + β π«π©πͺ = πππ therefore π¨π©πͺ is a straight?line. π΅ π΄ Bro Tip: Itβs a good idea to finish by stating the thing youβre trying to prove. π΅ πΆ π΄ π· π΅ π΄ πΆ Prove that the line π΄πΆ bisects β π΅π΄π·. β¦ β π©π¨πͺ = β πͺπ¨π« therefore π¨πͺ bisects ? β π©π¨π«. Prove that triangle π΄π΅πΆ is isosceles. β¦ β π©π¨πͺ = β π¨πͺπ© therefore ? π«πππ is isosceles. Exercises Question 1 [Set 4 Paper 1 Q4] π΄π΅πΆ is a right-angled triangle. Angle π΄πΆπ΅ = π₯. Angle π΅π΄π· = 90 β 2π₯. Prove that π΄πΆπ· is an isosceles triangle. ? Question 2 π΄π΅πΆπ· is a quadrilateral. Prove that π₯ = π¦. ? Question 3 π΄π΅ is parallel to πΆπ·. Is ππ parallel to ππ ? You must show your working. ? Question 4 πππ π is a cyclic quadrilateral. ππ = ππ . πππ is a tangent to the circle. Work out the value of π₯. You must show your working. ? Question 5 π΄, π΅, πΆ and π· are points on the circumference of a circle such that π΅π· is parallel to the tangent to the circle at π΄. Prove that π΄πΆ bisects angle π΅πΆπ·. Give reasons at each stage of your working. ? Question 6 Prove that π΄π΅ is parallel to π·πΆ. ? Question 7 π΄π΅πΆ is a triangle. π is a point on π΄π΅ such that π΄π = ππΆ = π΅πΆ. Angle π΅π΄πΆ = π₯. a) Prove that angle π΄π΅πΆ = 2π₯. b) You are also given that π΄π΅ = π΄πΆ. Work out the value of π₯. ?