A Very Nice Geometry Problem | You should be able to solve this! 

A = 80cm² = ½b.h = ½.2h.h = h² h = √80 = 4.√5 cm b = 2h = 8√5 cm tan α = ½x / x = 1/2 α = 26,565° x = b cos α x = 16 cm ( Solved √ )

BQ=a Triangles BPQ and PAD are similar ( angle BPQ= angle PDA). therfore, AP/BQ=AD/PB or X/2/a= X//X/2 or a=1/4X. Express all triangles areas in terms of X and make the sum=X^2. you will get an equation 16a^2+4a^2 -320=0. a=4. X=4a=16. ( no Phythagoras).

BQ=a..(√5x/2)^2+((x/2)^2+a^2)=x^2+(x-a)^2...a=x/4...(√5x/2)*√(5x^2/16)=160..(√5x/2)*(√5x/4)=160...5x^2=8*160..x^2=256..x=16

Here is my version with trigonometry. Angle ADP = angle BPQ =a tan(a)=AP/AD=(x/2)/x=1/2 from which 1+tan(a)^2=sec(a)^2 -->sec(a)=sqrt(5)/2 --cos(a)=2/sqrt(5) DP*cos(a)=x hence DP=x/cos(a) PQ*cos(a)=x/2 hence PQ=x/(2*cos(a)) Area(DPQ)=1/2*DP*PQ=1/2*x^2/(2*cos(a)^2)=1/2*x^2*5/8=80 X^2=256 hence x=16

x = 2L = side of the square. [Triangle] = a*b/2 where a=L*sqrt (5) and b = L*sqrt (5) /2 --> L^2 * 5/2/2 = 5 * L^2 / 4 = 80 --> L^2 = 80*4/5 = 64 --> L=8 --> x = 2L = 16

⊿DAP∞⊿PBQ AP=BP=x/2 BQ=x/4 QC=3x/4 DC=x ⊿DAP=x*x/2*1/2=x²/4 ⊿PBQ=x/2*x/4*1/2=x²/16 ⊿QCD=3x/4*x*1/2=3x²/8 ⊿DPQ=x²-(x²/4+x²/16+3x²/8)=5x²/16=80 5x²=1280 x²=256 x>0 , x=16

AP = 1/2 x DP= x√5/2 (by pythagoras) APD and PBQ are similar triangles By AA theorem So PQ/PB=DP/AD PQ=x/2*x√5/2x PQ=x√5/4 Area of triangle 80=5x^2/16 x^2=256 x= 16

360°ABCD/80x =40.40 2^20.2^20 1^5^4.1^5^4 1^2^2.1^2^2 1^1.1^2 1^2 (ABCDx ➖ 2ABCDx+1)

As ABCD is a square and AB = BC = CD = DA = x, AP = PB = x/2. Let ∠ADP = α, and ∠DPA = β, as ∠PAD = 90°, α and β are complementary angles that sum to 90°. As ∠QPD = 90°, ∠BPQ = 180°-(90°+ β) = 90°- β = α, so ∠PQB = β. ∆PAD and ∆QBP are therefore similar triangles by AAA. Triangle ∆PAD: PA² + AD² = DP² (x/2)² + x² = DP² DP² = x²/4 + x² = 5x²/4 DP = √(5x²/4) = √5x/2 Triangle ∆QBP PQ/BP = DP/AD PQ/(x/2) = (√5x/2)/x PQ = (x/2)(√5x/2)/x = √5x/4 Triangle ∆QPD: Aₜ = bh/2 = PD(QP)/2 80 = (√5x/2)(√5x/4)/2 80 = 5x²/16 x² = 80(16/5) = 16(16) x = 16 units

My Third elementary solution 180° rotation PBQ in P.....BQ=X/4=a Area BPQ a*2a/2 + Area DAP 2a*4a/2 = Area DPQ= 80 a^2+4a^2 = 5a^2 =80 a^2=16 a=4 X=4*a=4*4=16

φ = 30° → sin⁡(3φ) = 1; ∎ABCD → AB = AP + BP = a + a = x = BC = BQ + CQ = CD = AD = ? sin⁡(QPD) = 1 → (1/2)(QP)(DP) = 80; ADP = PDQ = BPQ = δ → sin⁡(δ) = √5/5 → cos⁡(δ) = 2√5/5 → tan⁡(δ) = sin⁡(δ)/cos⁡(δ) = 1/2 → PQ = a√5/2 → DP = a√5 → (1/2)(DP)(PQ) = 80 → a = 8 → 2a = x = 16

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PQ can be obtained as under..... angle BQP will be beta(B) PQ/PD = PB/AD Putting the values we will get PQ =√5.X/4

AD=X→ AP=X/2=PB→ PD²=X²+(X/2)²=5X²/4→ PD=X√5/2 → Razón de semejanza entre QBP y PAD: s=PB/AD=1/2→ PQ=X√5/4→ PD*PQ=2*80→ X=16. Gracias y un saludo cordial.

This explanation honestly *better* than all of the hearted comments. I hope that this means I can do this as a practice easily!!!

You could have found P by using similarity of triangles instead of doing the Pythagoras calculation again...

x=16

X=16

I did not read that abcd was a square in the original question. My bad

Here is an approach without using Pythagoras. Let _∠APD = θ_ In _ΔAPD: tanθ = 2_ ⇒ _tan(90° - θ) = ½_ In _ΔBPQ:_ _|BQ|/|BP| = |tan(∠BPQ)_ ⇒ _|BQ|/(½x) = tan(90° - θ) = ½_ ⇒ _|BQ| = (¼)x_ Area of _ΔAPD_ + Area of _ΔBPQ_ + Area of _ΔCDQ_ = Area of _□ABCD_ - Area of _ΔDPQ_ ⇒ _(¼)x² + (⅟₁₆)x² + (⅜)x² = x² - 80_ ⇒ _x² = 16²_ ⇒ *_x = 16_*

Pythagoras' theorem is not needed to solve this problem. ADP and BPQ are similar triangles so the angles ADP and BPQ are the same and you can calculate line BQ (which is x/4) Now you can calculate line CQ; Line CQ=x-x/4=3x/4 The square of x is equal to the four triangles inside it You can write this in a formula and calculate x ; x^2=1/2*((3x/4*x)+(x*x/2)+(x/2*x/4))+80 x^2-80=1/2*((3x^2/4)+(x^2/2)+(x^2/8)) 2*(x^2-80)=(6x^2/8)+(4x^2/8)+(x^2/8) 2x^2-160=(6x^2+4x^2+x^2)/8 2x^2-160=11x^2/8 8*(2x^2-160)=11x^2 16x^2-1280=11x^2 5x^2-1280=0 5x^2=1280 x^2=256 x=16

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X=16 cm

But 16√5 also correct