Very Nice Geometry Problem | You should be able to solve this | 2 Different Methods
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- Опубліковано 4 бер 2024
- Very Nice Geometry Problem | You should be able to solve this | 2 Different Methods
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@Math Boo. I liked too much the second solution !!!!! Another solution with areas :
Let α= side of square ABCD.
Draw the straight line segment EF and EG is the height of triangle DEF.
The triangles ADE=DEG , so EG=AE=5.
Area (ADE) + area (DEF) + area (BEF) + area (DFC) = area (ABCD) =>
½ •5α +½ • 5x + ½ •(α-5)(α-3) +1/2• 3α = α² ……….
5x+15= α² => 5x+15= x²-9 => x²-5x-24=0 cause α²=x²-9
……. x=8 cause x>0
Third method.
Join EF. COS theta = (x*2+DE*2-EF*2)/2X.DE
EF*2=(a-3)*2+(a-5)*2
DE*2=5*2+a*2
X*2=3*2+a*2
Solving above equations gives cos theta =8a/X.DE
From triangle DAE, cos theta=a/DE
equating both cos theta
8a/X.DE=a/DE
8/X=1
X=8
Let theta = "a" just for convenience. Then tan (a) = 5/L and tan (90-2a) = 3/L --> tan (2a) = L/3. Now tan (2a) = 2tan (a) / [1-(tan^2 (a))] -> L/3 = 2*5/L / [1-((5^2)/L^2)] = 10L / (L^2 - 25) -> L^2-25 = 3*10 = 30 -> L^2 = 55 -> X^2 = L^2 + 3^2 = 55+9 = 64 --> L=8 units
I used the first method, but solved for a without ever solving for tan θ
In △ADE, ∠DAE = 90°, ∠ADE = θ → tan θ = AE/AD = 5/a
In △CDF, ∠DCF = 90°, ∠CDF = 90−2θ → ∠DFC = 2θ → tan 2θ = CD/CF = a/3
Using Double Angle Identity, we get:
tan 2θ = 2 tan θ / (1 − tan²θ)
a/3 = (10/a) / (1 − 25/a²)
a/3 = 10a / (a² − 25)
1/3 = 10 / (a² − 25)
a² − 25 = 30
a² = 55
Using Pythagorean Theorem in △CDF, we get:
DF2 = CD2 + CF2
x² = a² + 3²
x² = 55 + 9 = 64
x = 8
At about 3:10,
My third solution in a few words .
Extend the line BA (to the left) and make segment GA=3
Obviously ∆ ADG = ∆ DFC => DG=DF=x
< GDA = < FDC =90-2θ
< GDE = x=8
If we assume that this is a square , not rectangle
we can solve it in that way
tan(theta) = 5/y
tan(pi/2-2theta) = 3/y
tan(theta) = 5/y
tan(2theta) = y/3
2*(5/y)/(1-25/y^2) = y/3
(10/y)/((y^2-25)/y^2) = y/3
10y/(y^2-25) = y/3
10y/(y^2-25) - y/3 = 0
y(10/(y^2-25) - 1/3)=0
y(10-1/3*(y^2-25)) = 0
30-y^2+25 = 0
y^2 = 55
From Pythagorean theorem
y^2+3^2=x^2
55+9 = x^2
x^2 = 64
x = 8
The second method is unbelievable!
equal angle opposite side is also equal.
let side of the square is y.
so, (y-5)^2+(y-3)^2=25 y= 7.39
so x = 7.98
X=8 i found by finding the side of square i produced DF to AB and used angle bister theorem i found the side =√55
So x=8
Master ,here is my solution
Super easy
We can use cosine theorem
in triangle ABE and EBF
to solve this problem.
I did the first method, but the second method is so much more interesting. I wish I had the math mind to think outside the box
5cot(θ)=s
3cot(π/2-2θ)=s
s/cos(π/2-2θ)=x
s=square side length.
^=read as square
Finally from the given facts we'll get a equation
X^2-10x+16=0
So
X=8 or 2
φ = 30°; AB = 5 + (5 - a) = BC = 3 + (a - 3) = CD = AD = a
ADE = EDF = θ; FDC = δ → 2θ + δ = 3φ → tan(δ) = cot(2θ) = 3/a → cot(δ) = tan(2θ) = a/3
tan(θ) = 5/a → tan(2θ) = 2tan(θ)/(1 - tan^2(θ)) = 10a/(a^2 - 55) = a/3 → a^2 = 55 → x = √(55 + 9) = 8 →
sin(δ) = 3/8 → EF = 4√(9 - √55)
... Good day, [ Let THETA = T ] ... (1) Triangle(ADE) .... TAN(T) = 5 / I AD I .... (2) Triangle(DCF) .... TAN(2T) = I DC I / 3 = I AD I / 3 ... TAN(2T) = 2 * TAN(T) / (1 - TAN^2(T)) .... so, I AD I / 3 = ( 2 * (5/I AD I) / ( 1 - [ 5/I AD I ]^2 ) ... after a few basic algebraic steps solving for I AD I or of course I DC I we obtain .... I AD I = I DC I = SQRT(55) .... finally computing the X- value in Triangle(DCF) by applying Pythagoras .... X^2 = 3^2 + (SQRT(55))^2 .... X = 8 ... thank you for your clear explanations sir ... best regards, Jan-W
let's construct a right-angled triangle ABC with angle theta at vertex C, opposite perpendicular BC of length 5.
Let's construct the point K on the perpendicular BC so that
|BK|=2.
Triangle ABK has
at vertex B angle 90-θ, opposite side x,
at vertex A angle 3θ-90, opposite side 2, applying Son's theorem we get the requirement
x=(2cosθ)/(-3cosθ)
Let's apply the function sin(90-2θ)=3/x to triangle AKC, i.e. x=3/cos2θ
We get a request for the corresponding x
x=2cosθ/(-cos3θ)=3/cos2θ
This will give the request
cosθ+cos3θ=-3cos3θ
so cosθ+4cos3θ=0
x=(2(cosθ+4cos3θ)-8cos3θ)/(-cos3θ)=(0-8cos3θ)/(-cos3θ)=8
Let Dxy an orthogonal axis system . D(0,0) , A(0,α) ,Ε(5,α) , F(α,3).
( α= side of square ABCD).
The vectors DA=(0,α) , DE=(5,α) , DF=(α,3) . Norm |DA|=α , |DF|=√( α² +9),
The scalar product of vectors : DA•DE = |DA|•|DE|•cosθ
DΕ•DF= |DE|•|DF|•cosθ
Deviding by term ; (DA•DE) / (DΕ•DF) = |DA|•/ |DF| =>
(0•5+ α² )/ (5•α +3•α) =α / √ ( α²+9)
We can find easily √( α²+9)=8 , so x=|DF|=√ (a² +9)=8
Alternatively, let side length of the square be a. Extend DE and CB, intersecting at G. DF=FG. x=FG=FB+BG=(a-3)+a(a-5)/5=(a^2-15)/5. x^2=a^2+3^2. a=sqrt(55), x=8
Very nice solution !!!!!!! Fix the equality FG=FB+FG in FG=FB + BG and all are fine .
@@Irtsak I have fixed the typo. Thanks!