Olympiad Math | Can you find the missing side lengths? |

Learn how to find the missing side lengths. Important Geometry and Algebra skills are also explained: Exterior angle theorem; congruent triangles; Pythagorean Theorem; right triangles; isosceles triangles. Step-by-step tutorial by PreMath.com
Today I will teach you tips and tricks to solve the given olympiad math question in a simple and easy way. Learn how to prepare for Math Olympiad fast!
Step-by-step tutorial by PreMath.com
• Olympiad Math | Can yo...
Need help with solving this Math Olympiad Question? You're in the right place!
I have over 20 years of experience teaching Mathematics at American schools, colleges, and universities. Learn more about me at
/ premath
Olympiad Math | Can you find the missing side lengths? | #math #maths | #geometry
Olympiad Mathematical Question! | Learn Tips how to solve Olympiad Question without hassle and anxiety!
#FindMissingSides #ExteriorAngleTheorem #CongruentTriangles #PythagoreanTheorem #RightTriangles #Triangle #AreaOfSquare #AreaOfTriangle #CircleTheorem #GeometryMath #EquilateralTriangle #PerpendicularBisectorTheorem
#MathOlympiad #ThalesTheorem #RightTriangle #RightTriangles #CongruentTriangles
#PreMath #PreMath.com #MathOlympics #HowToThinkOutsideTheBox #ThinkOutsideTheBox #HowToThinkOutsideTheBox? #FillInTheBoxes #GeometryMath #Geometry #RightTriangles
#OlympiadMathematicalQuestion #HowToSolveOlympiadQuestion #MathOlympiadQuestion #MathOlympiadQuestions #OlympiadQuestion #Olympiad #AlgebraReview #Algebra #Mathematics #Math #Maths #MathOlympiad #HarvardAdmissionQuestion
#MathOlympiadPreparation #LearntipstosolveOlympiadMathQuestionfast #OlympiadMathematicsCompetition #MathOlympics #CollegeEntranceExam
#blackpenredpen #MathOlympiadTraining #Olympiad Question #GeometrySkills #GeometryFormulas #Angles #Height
#MathematicalOlympiad #OlympiadMathematics #CompetitiveExams #CompetitiveExam
How to solve Olympiad Mathematical Question
How to prepare for Math Olympiad
How to Solve Olympiad Question
How to Solve international math olympiad questions
international math olympiad questions and solutions
international math olympiad questions and answers
olympiad mathematics competition
blackpenredpen
math olympics
olympiad exam
olympiad exam sample papers
math olympiad sample questions
math olympiada
British Math Olympiad
olympics math
olympics mathematics
olympics math activities
olympics math competition
Math Olympiad Training
How to win the International Math Olympiad | Po-Shen Loh and Lex Fridman
Po-Shen Loh and Lex Fridman
Number Theory
There is a ridiculously easy way to solve this Olympiad qualifier problem
This U.S. Olympiad Coach Has a Unique Approach to Math
The Map of Mathematics
mathcounts
math at work
Pre Math
Olympiad Mathematics
Two Methods to Solve System of Exponential of Equations
Olympiad Question
Find Area of the Shaded Triangle in a Rectangle
Geometry
Geometry math
Geometry skills
Right triangles
imo
Competitive Exams
Competitive Exam
Calculate the Radius
Equilateral Triangle
Pythagorean Theorem
Area of a circle
Area of the sector
Right triangles
Radius
Circle
Quarter circle
coolmath
my maths
mathpapa
mymaths
cymath
sumdog
multiplication
ixl math
deltamath
reflex math
math genie
math way
math for fun
Subscribe Now as the ultimate shots of Math doses are on their way to fill your minds with the knowledge and wisdom once again.

Пікірлер: 46

  • @MMmaths8800
    @MMmaths88002 ай бұрын

    Nice sir, I'm your big fan

  • @PreMath

    @PreMath

    2 ай бұрын

    Thanks dear❤️

  • @JLvatron
    @JLvatron2 ай бұрын

    Wow, amaze!

  • @dirklutz2818
    @dirklutz28182 ай бұрын

    You made it unbelievable simple

  • @robith100
    @robith1002 ай бұрын

    Let the height = h. h/8 = tan x h = 8.tan x h/2 = tan 2x h = 2.tan 2x We get 8.tanx = 2.tan 2x 4.tan x = tan 2x Knowing tan 2x = 2.tan x/(1-tan²x) So 4.tan x = 2.tan x/(1-tan²x) 1-tan²x = 2.tan x/4.tan x 1-tan²x = ½ 1-½ = tan²x ½ = tan²x tan x = ±√(1/2) = ±½√2 Since x is acute, then tan x = ½√2 h/8 = ½√2 h = 8.½√2 = 4√2 Using phytagorean for determining both hypotenuses, and we've done.

  • @PreMath

    @PreMath

    2 ай бұрын

    Excellent! Thanks for sharing ❤️

  • @robith100

    @robith100

    2 ай бұрын

    @@PreMath love ur channel 😍

  • @uwelinzbauer3973
    @uwelinzbauer39732 ай бұрын

    Hello! I fought hard, because I didn't see the much easier construction of auxiliary lines, I constructed an angle bisector from point B. Too complicated to describe, to type everything in here, but I found relations leading to three equations with 3 unknowns, finally and luckily found the values we were looking for - correctly. But your shown way was much easier and much faster. Why didn't I see that myself? I hope I will get better, but at least I am glad, that I did it myself and I didn't have to give up. BTW, I am speechless about how many videos you produce and put them on line. When I solved one, the next one is already waiting. My deep respect and thank you very much!

  • @ludmilaivanova1603
    @ludmilaivanova16032 ай бұрын

    use of formula - tan2A= 2tanA/1-tan^2A- makes it much easier.

  • @PreMath

    @PreMath

    2 ай бұрын

    Thanks for sharing ❤️

  • @GetMeThere1
    @GetMeThere12 ай бұрын

    AMAZING! Seeing some of your solutions is like seeing magic. Thank you so much for providing this wonderment!

  • @ChuzzleFriends
    @ChuzzleFriends2 ай бұрын

    By definition of acute angles in a right triangle, m∠ACD = (90 - x)°. Draw a segment thru point C and a point E on segment AD, such that △BCE is an isosceles triangle. So, BC = CE, BD = DE = 2, & m∠BEC = (2x)° by the Base Angles Theorem. Then, AE = 8 - 2 = 6 by the Segment Addition Postulate. Again, by definition of acute angles in a right triangle, m∠DCE = (90 - 2x)°. Use the Angle Addition Postulate. m∠ACD = m∠ACE + m∠DCE (90 - x)° = m∠ACE + (90 - 2x)° m∠ACE = (90 - x)° - (90 - 2x)° = 90 - x - 90 + 2x m∠ACE = x° Therefore, △AEC is an isosceles triangle and AE = CE by the Base Angles Converse. By the Transitive Property of Equality (or Segment Congruence), BC = 6. One down, two to go! Apply the Pythagorean Theorem on △BDC (or △EDC). a² + b² = c² (CD)² + 2² = 6² (CD)² + 4 = 36 (CD)² = 32 CD = √32 = (√16)(√2) = 4√2 For the last side, use the Pythagorean Theorem on △ADC. c² = a² + b² (AC)² = (4√2)² + 8² = 32 + 64 = 96 AC = √96 = (√16)(√6) = 4√6 Thus, the lengths of the missing sides are as follows: AC = 4√6 units ≈ 9.80 units BC = 6 units CD = 4√2 units ≈ 5.66 units

  • @unknownidentity2846
    @unknownidentity28462 ай бұрын

    Let's do it: . .. ... .... ..... Since the triangles ACD and BCD are right triangles, we can conclude: tan(x) = CD/AD ∧ tan(2x) = CD/BD ⇒ tan(2x)/tan(x) = (CD/BD)/(CD/AD) = (CD/BD)*(AD/CD) = AD/BD = 8/2 = 4 tan(2x) = 4*tan(x) 2*tan(x)/[1 − tan²(x)] = 4*tan(x) 1/2 = 1 − tan²(x) tan²(x) = 1/2 ⇒ tan(x) = 1/√2 = √2/2 CD/AD = tan(x) ⇒ CD = tan(x)*AD = (√2/2)*8 = 4√2 Now we can apply the Pythagorean theorem to the right triangles ACD and BCD: AC² = AD² + CD² = 8² + (4√2)² = 64 + 32 = 96 ⇒ AC = √96 = 4√6 BC² = BD² + CD² = 2² + (4√2)² = 4 + 32 = 36 ⇒ BC = √36 = 6 Best regards from Germany

  • @PreMath

    @PreMath

    2 ай бұрын

    Excellent! Thanks for sharing ❤️

  • @wackojacko3962
    @wackojacko39622 ай бұрын

    Refreshing! 🙂

  • @PreMath

    @PreMath

    2 ай бұрын

    Glad to hear that! Thanks ❤️

  • @prossvay8744
    @prossvay87442 ай бұрын

    Tan(x)=h/8 CD=h Tan(2x)=h/2 Tan(2x)=2tan(x)/1-tan(x)^2=h/2 2 h/8/1-(h/8)^2=h/2 So h=4√2 AC=√64+(4√2)^2=4√6 units CD=h=4√2 units BC=√2^2+(4√2)^2=6 units.❤❤❤ Thanks sir.

  • @PreMath

    @PreMath

    2 ай бұрын

    Excellent! You are very welcome! Thanks for sharing ❤️

  • @jamestalbott4499
    @jamestalbott44992 ай бұрын

    Nice!

  • @CloudBushyMath
    @CloudBushyMath2 ай бұрын

    Superb!👍

  • @PreMath

    @PreMath

    2 ай бұрын

    Excellent! Thanks ❤️

  • @quigonkenny
    @quigonkenny2 ай бұрын

    Draw CE, where E is the point on AD where ED = 2. As ED = DB, ∠EDC = ∠CDB, and CD is shared, ∆EDC and ∆CDB are congruent by SAS and thus ∠CED = ∠DBC = 2x. As ∠CED is an external angle to ∆AEC at E, ∠ECA = ∠CED - ∠CAE = 2x-x = x. As ∠ECA = ∠CAE = x, ∆AEC is isosceles and EC = AE = 8-2 = 6. As ∠EBC = ∠CEB = 2x, ∆BCE is isosceles and BC = CE = 6. BC = 6 ✓ Triangle ∆CDB: DB² + CD² = BC² 2² + CD² = 6² CD² = 36 - 4 = 32 CD = √32 = 4√2 CD = 4√2 ✓ Triangle ∆ADC: DC² + AD² = CA² (4√2)² + 8² = CA² CA² = 32 + 64 = 96 CA = √96 = 4√6 CA = 4√6 ✓

  • @laxmikantbondre338
    @laxmikantbondre3382 ай бұрын

    Nice Solution. But small correction the 2 triangles are congruent by AAS not by ASA.

  • @johnbrennan3372
    @johnbrennan33722 ай бұрын

    Using tan x and tan 2x and equating the two values for cd works out easy giving tan x = 1/sqroot2. So cd/ 8= 1/sqroot2 etc

  • @PreMath

    @PreMath

    2 ай бұрын

    Thanks for sharing ❤️

  • @MateusMuila
    @MateusMuila2 ай бұрын

    this one was easy thanks to your amazing explanation , I really appreciate . I suggest also calculus of Maclaurin and Taylor series if possible . thanks to all your video lessons I understand many stuffs in geometry . Thanks Sir

  • @user-ve6wo2gd5n
    @user-ve6wo2gd5n2 ай бұрын

    I really appreciate your video lessons, it really make my days happier, Could you mind Telling us your name. I express my sincere gratitude to you, Sir.

  • @luigipirandello5919
    @luigipirandello59192 ай бұрын

    Very nice solution. Thank you.

  • @PreMath

    @PreMath

    2 ай бұрын

    You are very welcome! Thanks ❤️

  • @Irishfan
    @Irishfan2 ай бұрын

    I like to see answers in decimal form. CB = 5.6569 AC = 9.7980 note I show equals sign because my tablet doesn't have an approximate sign. I carry them out to the nearest ten thousandth just for further use in calculations. The work l do in construction and civil engineering usually doesn't require final answers in a degree of accuracy that high. When I do calculations using a calculator, I leave the preliminary answers in memory carried out as far as the calculator will display. When I look at a number that is 5.66 units I know how long that is, when I look at 4 x the square root of 2, I have no idea how long that is and I have to get out my calculator convert it to decimal form. So, to me, when a math instructor wants an answer simplified, the simplest form wouldn't be an answer with a square root in it. The simplest form would be a digital number.

  • @johnpiggott7426
    @johnpiggott74262 ай бұрын

    I got two answers, [yellow, middle, blue] =[4sqrt10, 4sqrt6, 10] or [4sqrt6, 4sqrt2, 6]. They check when a2+b2=c2 for the yellow and blue triangles, but the angles dont check for the first set, so the second set is correct.

  • @DB-lg5sq
    @DB-lg5sq2 ай бұрын

    شكرا لكم على المجهودات يمكن استعمال tanx=h/8 tan2x=h/2 ...... h=4 (racine 2)

  • @LuisdeBritoCamacho
    @LuisdeBritoCamacho2 ай бұрын

    1) tan(X) = CD/8 2) tan(2X) = CD/2 3) CD = 8*tan(X) 4) CD = 2*tan(2X) 5) 8*tan(X) = 2*tan(2X) 6) tan(2X) = 2*tan(X) / (1 − tan^2(X)) 7) CD = h 8) h = 8*tan(X) 9) h = 2*tan(2X) 10) 8*tan(X) = 2*tan(2X) 11) 8*tan(X) = 4*(tan(X) / (1 - tan^2(X)) 12) 8*(1 - tan^2(X) = 4 13) 1 - tan^2(X) = 1/2 14) 1 - (h/8)^2 = 1/2 15) 1 - h^2/64 = 1/2 16) 64 - h^2 / 64 = 1/2 17) 64 - h^2 = 32 18) h^2 = 64 - 32 19) h^2 = 32 20) h = sqrt(32) = 4*sqrt(2) ~ 5,7 lin un 21) X = arctan(4*sqrt(2)/8) ; X = arctan(sqrt(2)/2) ; X ~ 35,3º 22) 2X = arctan(4*sqrt(2)/2) ; 2X = arctan(2*sqrt(2)) ; 70,5º 23) AC^2 = 64 + 32 ; AC^2 = 96 ; AC = sqrt(96) ; AC = 4*sqrt(6) ; AC ~ 9,8 lin un 24) BC^2 = 4 + 32 ;BC^2 = 36 ; BC = 6 lin un 25) My Best Answer is that AC is equal to 4*sqrt(6) Linear Units (approx. 9,8 Linear Units) and BC = 6 Linear Units.

  • @PreMath

    @PreMath

    2 ай бұрын

    Excellent! Thanks for sharing ❤️

  • @marcgriselhubert3915
    @marcgriselhubert39152 ай бұрын

    In triangle ADC: tan(x) = DC/8, and in triangle BDC: tan(2.x) = DC/2, so we have tan(2.x) = 4.tan(x). Or tan(2.x) = (2.tan(x))/ (1 -tan(x)^2) So 4.tan(x) = (2.tan(x))/(1 -tan(x)^2) and 4.tan(x) - 4.tan(x)^3 = 2.tan(x), that gives tan(x)^2 = 1/2 and tan(x) = sqrt(2)/2 as tan(x) >0 Now we get sqrt(2)/2 = DC/8 and DC = 4.sqrt(2). Then in triangle ADC: AC^2 = 8^2 + (4.sqrt(2))^2 = 96 and AC = sqrt(96) = 4.sqrt(6) and in triangle BDC: BC^2 = 2^2 + (4.sqrt(2))^2 = 36 and BC = sqrt(36) = 6.

  • @misterenter-iz7rz
    @misterenter-iz7rz2 ай бұрын

    a/2=tan 2x, a/8=tan x, so 4=2/(1-tan^2 x), tan^2 x=1/2, a=8 tan x=8/sqrt(2), sec^2 x=3/2, cos^2 x=2/3, sin^2 x=1/3, b cos x=8, b=8/cos x=8sqrt(3)/sqrt(2), cos 2x=2×(2/3)-1=1/3, c cos 2x=2, c=2/cos 2x=2/(1/3)=6.

  • @PreMath

    @PreMath

    2 ай бұрын

    Excellent! Thanks for sharing ❤️

  • @adept7474
    @adept74742 ай бұрын

    Much easier: bisector ВК ∠СВD. ▲АСD ~ ▲ВDК. DК/КС = 1/3. ВD/ВС = 1/3 (bisector property). ВС = 6. СD = 4√2, АС = 4√6 (Pythagorean theorem).

  • @PreMath

    @PreMath

    2 ай бұрын

    Thanks for sharing ❤️

  • @MegaSuperEnrique
    @MegaSuperEnrique2 ай бұрын

    1:37 AAS congruency, not ASA

  • @rabotaakk-nw9nm

    @rabotaakk-nw9nm

    2 ай бұрын

    1:10 construct: ED=BD=2, not SAS congruence =>

  • @user-jd5uz5xr9u
    @user-jd5uz5xr9u2 ай бұрын

    Pice of a cake. It is about inner and outer engels in a circle.

  • @soli9mana-soli4953
    @soli9mana-soli49532 ай бұрын

    It can be easily solved isoscelizing....inside or outside 😂

  • @Asphalt888.8
    @Asphalt888.82 ай бұрын

    2nd comment

  • @PreMath

    @PreMath

    2 ай бұрын

    Thanks ❤️

  • @murdock5537
    @murdock55372 ай бұрын

    Nice! φ = 30°; ∆ ABC → AB = AD + BD = 8 + 2 = 10; BC = a = ?; AC = b = ?; CD = h = ? sin⁡(ADC) = 1 → a = √(h^2 + 4); b = √(h^2 + 64); sin⁡(2δ) = h/a; sin⁡(δ) = h/b → cos⁡(δ) = 8/b → sin⁡(2δ) = 2sin⁡(δ)cos⁡(δ) → 16/b^2 = 1/a → 16a = b^2 → h^2 = a^2 - 4 = 16a - 64 → (a - 8)^2 = 4 → a1 = 10 = AB; a2 = 6 = a → h = 4√2 → b = 4√6 → sin⁡(δ) = √3/3 → δ ≈ 35,26°

Келесі