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1998 JBMO Problems/Problem 2: Difference between revisions

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Let <math>BC = a, ED = 1 - a</math>
Let <math>BC = a, ED = 1 - a</math>


Let angle <math>DAC</math> = <math>X</math>
Let <math>\angle DAC = X</math>


Applying cosine rule to triangle <math>DAC</math> we get:
Applying cosine rule to <math>\triangle DAC</math> we get:


<math>Cos X = (AC ^ {2} + AD ^ {2} - DC ^ {2}) / (2 * AC * AD )</math>
<math>\cos X = \frac{AC ^ {2} + AD ^ {2} - DC ^ {2}}{ 2 \cdot AC \cdot AD }</math>


Substituting <math>AC^{2} = 1^{2} + a^{2}, AD ^ {2} = 1^{2} + (1-a)^{2}, DC = 1</math> we get:
Substituting <math>AC^{2} = 1^{2} + a^{2}, AD ^ {2} = 1^{2} + (1-a)^{2}, DC = 1</math> we get:


<math>Cos^{2} X = (1 - a - a ^ {2}) ^ {2} / ((1 + a^{2})(2 - 2a + a^{2}))</math>
<math>\cos^{2} X = \frac{(1 - a - a ^ {2}) ^ {2}}{((1 + a^{2})(2 - 2a + a^{2}))}</math>


From above, <math>Sin^{2} X = 1 - Cos^{2} X  =  1 / ((1 + a^{2})(2 - 2a + a^{2})) = 1/(AC^{2}.AD^{2})</math>
From above, <math>\sin^{2} X = 1 - \cos^{2} X  =  \frac{1}{((1 + a^{2})(2 - 2a + a^{2}))} = \frac{1}{AC^{2} \cdot AD^{2}}</math>


Thus,  <math>Sin X * AC * AD = 1</math>
Thus,  <math>\sin X \cdot AC \cdot AD = 1</math>


So, <math>Area</math> of triangle <math>DAC</math> = <math>(1/2)*Sin X * AC * AD = 1/2</math>
So, area of <math>\triangle DAC</math> = <math>\frac{1}{2}\cdot \sin X \cdot AC \cdot AD = \frac{1}{2}</math>


Let <math>AF</math> be the altitude of triangle DAC from A.  
Let <math>AF</math> be the altitude of <math>\triangle DAC</math> from <math>A</math>.  


So <math>1/2*DC*AF = 1/2</math>
So <math>\frac{1}{2}\cdot DC\cdot AF = \frac{1}{2}</math>


This implies <math>AF = 1</math>.  
This implies <math>AF = 1</math>.  


Since <math>AFCB</math> is a cyclic quadrilateral with <math>AB = AF</math>, traingle <math>ABC</math> is congruent to <math>AFC</math>.  
Since <math>AFCB</math> is a cyclic quadrilateral with <math>AB = AF</math>, <math>\triangle ABC</math> is congruent to <math>\triangle AFC</math>.  
Similarly <math>AEDF</math> is a cyclic quadrilateral and traingle <math>AED</math> is congruent to <math>AFD</math>.  
Similarly <math>AEDF</math> is a cyclic quadrilateral and <math>\triangle AED</math> is congruent to <math>\triangle AFD</math>.  


So <math>area</math> of triangle <math>ABC</math> + <math>area</math> of triangle <math>AED</math> = <math>area</math> of Triangle <math>ADC</math>.
So area of <math>\triangle ABC</math> + area of <math>\triangle AED</math> = area of <math>\triangle ADC</math>.
Thus <math>area</math> of pentagon <math>ABCD</math> = <math>area</math> of <math>ABC</math> + <math>area</math> of <math>AED</math> + <math>area</math> of <math>DAC</math> = <math>1/2 + 1/2 = 1</math>
Thus area of pentagon <math>ABCD</math> = area of <math>\triangle ABC</math> + area of <math>\triangle AED</math> + area of <math>\triangle DAC</math> = <math>\frac{1}{2}+\frac{1}{2} = 1</math>






By <math>Kris17</math>
By <math>Kris17</math>


=== Solution 2 ===
=== Solution 2 ===

Revision as of 22:23, 4 June 2020

Problem 2

Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.


Solutions

Solution 1

Let $BC = a, ED = 1 - a$

Let $\angle DAC = X$

Applying cosine rule to $\triangle DAC$ we get:

$\cos X = \frac{AC ^ {2} + AD ^ {2} - DC ^ {2}}{ 2 \cdot AC \cdot AD }$

Substituting $AC^{2} = 1^{2} + a^{2}, AD ^ {2} = 1^{2} + (1-a)^{2}, DC = 1$ we get:

$\cos^{2} X = \frac{(1 - a - a ^ {2}) ^ {2}}{((1 + a^{2})(2 - 2a + a^{2}))}$

From above, $\sin^{2} X = 1 - \cos^{2} X = \frac{1}{((1 + a^{2})(2 - 2a + a^{2}))} = \frac{1}{AC^{2} \cdot AD^{2}}$

Thus, $\sin X \cdot AC \cdot AD = 1$

So, area of $\triangle DAC$ = $\frac{1}{2}\cdot \sin X \cdot AC \cdot AD = \frac{1}{2}$

Let $AF$ be the altitude of $\triangle DAC$ from $A$.

So $\frac{1}{2}\cdot DC\cdot AF = \frac{1}{2}$

This implies $AF = 1$.

Since $AFCB$ is a cyclic quadrilateral with $AB = AF$, $\triangle ABC$ is congruent to $\triangle AFC$. Similarly $AEDF$ is a cyclic quadrilateral and $\triangle AED$ is congruent to $\triangle AFD$.

So area of $\triangle ABC$ + area of $\triangle AED$ = area of $\triangle ADC$. Thus area of pentagon $ABCD$ = area of $\triangle ABC$ + area of $\triangle AED$ + area of $\triangle DAC$ = $\frac{1}{2}+\frac{1}{2} = 1$


By $Kris17$

Solution 2

Let $BC = x, DE = y$. Denote the area of $\triangle XYZ$ by $[XYZ]$.

$[ABC]+[AED]=\frac{1}{2}(x+y)=\frac{1}{2}$

$[ACD]$ can be found by Heron's formula.

$AC=\sqrt{x^2+1}$

$AD=\sqrt{y^2+1}$

Let $AC=b, AD=c$.

\begin{align*} [ACD]&=\frac{1}{4}\sqrt{(1+b+c)(-1+b+c)(1-b+c)(1+b-c)}\\ &=\frac{1}{4}\sqrt{((b+c)^2-1)(1-(b-c)^2)}\\ &=\frac{1}{4}\sqrt{(b+c)^2+(b-c)^2-(b^2-c^2)^2-1}\\ &=\frac{1}{4}\sqrt{2(b^2+c^2)-(b^2-c^2)^2-1}\\ &=\frac{1}{4}\sqrt{2(x^2+y^2+2)-(x^2-y^2)^2-1}\\ &=\frac{1}{4}\sqrt{2((x+y)^2-2xy+2)-(x+y)^2(x-y)^2-1}\\ &=\frac{1}{4}\sqrt{5-4xy-(x-y)^2}\\ &=\frac{1}{4}\sqrt{5-(x+y)^2}\\ &=\frac{1}{2} \end{align*}

Total area $=[ABC]+[AED]+[ACD]=\frac{1}{2}+\frac{1}{2}=1$.

By durianice

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