Olympiad Math | Calculate Area of the Blue shaded Triangle | Important Geometry skills explained

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!

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

Olympiad Math | Calculate Area of the Blue shaded Triangle | Important Geometry skills explained

Olympiad Mathematical Question! | Learn Tips how to solve Olympiad Question without hassle and anxiety!

#FindAreaOfTheTriangle #AreaOfTriangle #GeometryMath #PythagoreanTheorem
#MathOlympiad #AngleBisectorTheorem
#OlympiadMathematicalQuestion #HowToSolveOlympiadQuestion #MathOlympiadQuestion #MathOlympiadQuestions #OlympiadQuestion #Olympiad #AlgebraReview #Algebra #Mathematics #Math #Maths #MathOlympiad #HarvardAdmissionQuestion
#MathOlympiadPreparation #LearntipstosolveOlympiadMathQuestionfast #OlympiadMathematicsCompetition #MathOlympics #ExteriorAngleTheorem #PythagoreanTheorem #IsoscelesTriangles #AreaOfTriangleFormula #AuxiliaryLines
#blackpenredpen #MathOlympiadTraining #Olympiad Question #GeometrySkills #GeometryFormulas #AreaOfTriangles #CompetitiveExams #CompetitiveExam
#MathematicalOlympiad #OlympiadMathematics #LinearFunction #TrigonometricRatios

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
Olympiad Question
Find Area of the Shaded Triangle
Geometry
Geometry math
Geometry skills
Right triangles
Exterior Angle Theorem
pythagorean theorem
Isosceles Triangles
Area of the Triangle Formula
Competitive exams
Competitive exam
Find Area of the Triangle without Trigonometry
Find Area of the Triangle
Auxiliary lines method
Pythagorean Theorem
Square
Diagonal
Congruent Triangles
Angle Bisector Theorem

Subscribe Now as the ultimate shots of Math doses are on their way to fill your minds with the knowledge and wisdom once again.

[rishudubey]

thanks a lot dear professor 😊
i like your all vdos 😊your vdos make my day 😊

[zsoltszigeti]

tg 2a=2*tg a/(1-tg^2 a)
18/b=2*(8/b)/(1-8^2/b^2) => b=24 => A1=8*24/2=96, A=18*24/2=216, A2=A-A1=216-96=120

[theoyanto]

Hallelujah, ok I did it the the long way, but I dropped a perpendicular, you'll know where I mean, then spotted the reflection, from then on I was able to derive all I needed. This is the first time I've successfully expressed a variable in terms of another and I am so very grateful for your efforts in getting me to this point. I'm beginning to observe more than I thought I ever could, thank you. Still got much to learn, but your work is improving my mind immeasurably, not only in geometry, but in other areas of life.
But your use of "k" is just a top class act.
Thanks again 👍🏻

[matthewleitch]

This theorem is not taught in schools in the UK, as far as I know, but I knew about it from my mother's 1952 O level exams. I also solved this using Tan of the angle and Tan of twice the angle, then solving the equations to find tan of the angle, and hence the length of the base line.

[seleneendymion]

If we lower it perpendicular to the side bc from point d, its length will be 8 and the resulting triangle of 6 8 10 will be similar to the large triangle. therefore, since the height of the blue triangle is 8 and its base is 30, we find the area 120
( Hello from Turkey :) )

[harikatragadda]

Awesome! Another way is to Reflect Triangle DAB about the line DB so that DA meets perpendicularly on BC at A'. Since DA' is 8 and DC = 10 in the Right Triangle DA'C,
we have CA' = 6.
Since A'B = b, by Pythagoras theorem,
18² + b² = (6+b)²
b = 24

[wackojacko]

Reflection of Triangle DAB about Line DB is the coolest way to go...but I absolutely love the first PreMath solution.!🙂

[SladeMacGregor]

Thank you Professor PreMath! Wonderful, Awesome problem. Very interesting.

[JSSTyger]

I just wanted to say that I am well beyond my high school (and college) years but I enjoy these problems and I do them to keep my mind sharp. What you are doing is great and I hope you keep churning them out. As for this problem, my answer is A = 120. I get that from flipping triangle ADB about the line DB and noticing that the perpendicular distance from point D to the opposite side is 8. This allows us to solve some smaller triangles and then the overall triangle by using similar triangles. The area of the blue region then quickly follows.

[Ramkabharosa]

Let x= |AB| & θ=∠ABD. Then ∠ABC=2θ. So tan(θ)=8/x & tan(2θ)=18/x.
Hence 18/x = tan(2θ) = 2.tan(θ) / {1 - tan²(θ)} = (16/x) / {1 - 64/x²}.
So {1- 64/x²} = (16/x)/(18/x)= 8/9= {1-1/9}. ∴ 64/x²=1/9. So x²=9.(64).
So |AB|= x= 3.(8)=24 & |BC|= √(18²+24²)= √{6².(3²+4²)}= √{6².(5²)}= 30.
Area(ΔBCD)= (½).|CD|.|AB|= (½).(10).(24)= 120. Trigonometry is KING!

[ybodoN]

Generalized : the area of the blue triangle is _½ cd √((c + d) / (c − d))_ where _c_ = CD and _d_ = AD

[MrMichelX]

Thanks for this first theorem I didn't know. May be useful.

[bekaluu]

Used double angle identity tab2x=2tanx/(1-tan^2x) to calculate the unknown leg of the given right triangle. The rest is piece of cake. Thanks

[Irishfan]

I found the sides of this in a simpler way. I added 8+10 and got 18. I realized that this was a multiple of 3 and that this being a right triangle that the other two sides are going to be multiples of 4 and 5. So then, dividing 18 by 3, I got 6. Then, using 6 as the multiplier, I got 24 and 30 for the other sides of the triangle. From there, I found the area as in the video.

[manojitmaity]

How did you develop this formula ???
Please let me know.

[rmela]

Boy did i do this more complicatedly. I called the two angles alpha and the base x. Used pythagorean thm to express the two hypotenuses. Defined cos(alpha) two ways, first using the bottom triangle. Then used cosine law with upper triangle. Used substitution and it magically worked out nicely.

[jackrubin]

I would just drop a perpendicular line along line BC to intersect at D. Line DE must be equal to 8 as triangles BAD and BED are identical. Therefore area of triangle CDB = 0.5*8*30 = 120 square units.

[williamwingo]

Call the bottom line of the triangle(s) x, and let each of the equal angles be a. Then
tan a = 8/x; and tan (2a) = 18/x; but by the double-angle formulae, tan (2x) = 2 tan x / (1 – tan^2 (x)) so,
tan (2a) = 18/x = (2)(8/x) / (1 – (8/x)^2) = (16/x) / ((1 – (64/x^2)); and then,
18/x = (16/x) / ((1 – (64/x^2)); take a common denominator in the right:
18/x = (16/x) / ((x^2 – 64)/x^2); invert the divisor and multiply:
18/x = (16/x) (x^2/x^2 – 64) simplify and collect terms:
18/x = 16x/(x^2 – 64);
18x – ((64)(18)/x) = 16/x;
2x = ((64)(18)/x);
x = ((32)(18)/x);
x^2 = (32)(18) (16)(2)(9)(2) = (4^2)(3^2)(2^2);
so x = (4)(3)(2) = 24.
so the blue triangle has base 24 and altitude 10; and the area is (½)(24)(10) = 120.
Cheers. 🤠

[kunalchakraborty]

It can be also be solved by this method,
Let angle B=2*a,
Tan 2a=2tan a/1-tan^2 a
Tan 2a=18/b
Tan a=8/b
So
Putting the values of tan 2a and tan a
2 * 8/b/ 1- 64/b^2 = 18/b
Where b is the Base, after solving the value of b is found to be 24
So area= 1/2 * 10* 24=120

[DB-lgsq]

شكرا لكم
يمكن استعمال Hالمسقط العمودي لDعلى(BC)
نجد DH=AD
المثلث DHCقاءم الزاوية في H نجد CH=6 باستعمال cos الزاوية DCH ايACB نجدBH=24 اذن المساحة المطلوبة هي 120 = 2 /30×8