The natural numbers isosceles triangle percentages

Stella Seremetaki Key Stage 3,4Natural numbers –isosceles triangle percentages www.mathschool-online.com Train to …..learn maths

Introduction ................................................................................................. 2Greek mathematics ....................................................................................... 2Revision themes, Key Stage 3,4 .................................................................... 3Algebra .......................................................................................................... 3Question 1...................................................................................................... 3a) Two numbers a, b are called prime among them when theirmaximum common divisor is the unit....................................................... 3Question 2...................................................................................................... 3Answear 2. ..................................................................................................... 3Question 3...................................................................................................... 4Answear 3. ..................................................................................................... 4Geometry ........................................................................................................ 4Question 1...................................................................................................... 4When do two corners called vertica? .......................................................... 4Answear 1. ..................................................................................................... 4Answear 1. ..................................................................................................... 5Question 2...................................................................................................... 5Answear 2. ..................................................................................................... 6Exercise 1........................................................................................................ 6Calculate the values of the following scenarios ........................................ 6Answear 1. ..................................................................................................... 7Exercise 2........................................................................................................ 7Proof ............................................................................................................... 8Exercise 3........................................................................................................ 7Answer3. ......................................................................................................... 7Thank You!..................................................................................................... 8The paraboles ................................................................................................ 8y=ax2 , a>0 ..................................................................................................... 8Coming soon!................................................................................................. 8 1

Introduction Greek mathematicsGreek mathematics refers to mathematics texts andadvances written in Greek, developed from the 7thcentury BC to the 4th century AD around the shores ofthe Eastern Mediterranean. Greek mathematicianslived in cities spread over the entire EasternMediterranean from Italy to North Africa but wereunited by culture and language. Greek mathematics ofthe period following Alexander the Great is sometimescalled Hellenistic mathematics. The word\"mathematics\" itself derives from the Ancient Greek:μάθημα, translit. máthēma Attic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning \"subject ofinstruction\" The study of mathematics for its own sakeand the use of generalized mathematical theories andproofs is the key difference between Greek mathematicsand those of preceding civilizations.Euclid is the anglicized version of the Greek nameΕυκλείδης which means \"renowned, glorious\".https://en.wikipedia.org/wiki/Euclid 2

Revision themes, Key Stage 3,4 Algebra Question 1. Which numbers are called prime? Give an example Answear 1. a) Two numbers a, b are called prime among them when their maximum common divisor is the unit. ΜCD(α,β)=1 For example the numbers 2,7 are primes ΜCD(2,7)=1 Question 2.What we call a minimum common multiple of two or more numbers? Answear 2.b) A minimum common multiple of two or more numbers is called thesmallest of their common multiples 3

Question 3.Mark the following suggestions withTrue or False Answear 3. i) T, the number 2010 ends at a zero, therefore it is divided by 10. ii) F, the number 1453 is not dividedby 3 because the sum of its digits, ie 1 + 4 + 5 + 3 = 13 is not divided by 3(ii) T, the number 1821 is divided by 1because each number is divided by the unit Geometry Question 1. When do two corners called vertica? Answear 1. a) Two angles ω1 and ω2 are called verticals when they have a common vertex Ο and their sides are oblique. 4

b) When do two angles called complementary?Find the complementary angle θ = 60 degrees Answear 1. b) Two corners are called complementary when they have a sum of 180 degrees. The complementary angle θ = 60 degrees is the angle of 120 degrees Indeed: 1200+θ=1200+600=1800 Question 2.Assign each corner of column A with its complementary column B 5

Column Α Column Βi)300 c)300ii)450 b)450iii)600 a)600www.mathschool-online.com Answear 2.c) To match every corner of column A with its complementary column B, I must knowthat if they are complementary they have a sum of 900Column Α Column Βi)300 c)600ii)450 b)450iii)600 a)300www.mathschool-online.com Exercise 1.Calculate the values of the following scenarios a) Α=(+3)+(-4)+(+1) b) B=(+4/6)-(-3/6)+(-12/6) 6

Answear 1. a) Α=(+3)+(-4)+(+1)=(-1)+(+1)=0 b) B=(+4/6)-(-3/6)+(-12/6) I find the minimum common multiple of the denominators to convert fractions to homonymous MCM=6B=(+4/6)-(-3/6)+(-12/6)=4/6 + 3/6 - 12/6= 7/6 - 12/6 = -5/6 Exercise 2.Ηypothesis Ιnduction ΑΒ//ΕΔ Proof that the ABGβ=450,γ=450 triangle is right and isosceleswww.mathschool-online.com 7

Proof a) I know from the hypothesis that β = 45degrees I know from the hypothesis that ΑΒ//ΕΔ and that the AB and EΔ lines are intersected by a third straight EB This means that the alternating angles formed are equal.Thus angle B equals the angle A and is 45 degrees, so the triangle ABC is isoscelesI know that in each triangle ABC the sumof its angles is 180 degrees, so the triangle ABC is rectangular to C, so the triangle ABC is rectangular and isosceles 8

Exercise 3.Students of the First High School competedin mathematics. If we know that 1/4 wrote under the base and 45 students wrote above the base, find (a) how many students were in the first grade? (b) what percentage he wrote down the base? Answer3. Let X be the total of the students of the First High School a) We know that 1/4 of the students wrote down the base. This means that (������ / 4) x students wrote under the base,while the 45 students wrote above the base Therefore:(Total of students) - (45 above the base) = (students who wrote below the base) 7

Equivalentx - 45 = ������ x→ ������ I am deleting the equation to x.I turn the unknown to Part 1 and the familiar to part 2 Thereforex= 60 studentsb) The percentage of students who wroteabove the base is calculated as follows: (45/60).100=0,75.100=75% Thank You!The parabolesy=ax2 , a>0Coming soon! 8