Difference between revisions of "Wooga Looga Theorem"

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==Application 1==
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===Problem 1===
 
The Wooga Looga Theorem states that the solution to this problem by franzliszt:
 
The Wooga Looga Theorem states that the solution to this problem by franzliszt:
  
 
In <math>\triangle ABC</math> points <math>X,Y,Z</math> are on sides <math>BC,CA,AB</math> such that <math>\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71</math>. Find the ratio of <math>[XYZ]</math> to <math>[ABC]</math>.
 
In <math>\triangle ABC</math> points <math>X,Y,Z</math> are on sides <math>BC,CA,AB</math> such that <math>\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71</math>. Find the ratio of <math>[XYZ]</math> to <math>[ABC]</math>.
 
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===Solution 1===
 
is this solution by RedFireTruck:
 
is this solution by RedFireTruck:
  
 
WLOG let <math>A=(0, 0)</math>, <math>B=(1, 0)</math>, <math>C=(x, y)</math>. Then <math>[ABC]=\frac12|y|</math> and <math>X=(\frac{7x+1}{8}, \frac{7y}{8})</math>, <math>Y=(\frac{x}{8}, \frac{y}{8})</math>, <math>Z=(\frac78, 0)</math>. Then <math>[XYZ]=\frac12|\frac{43y}{64}|</math>. Therefore the answer is <math>\boxed{\frac{43}{64}}</math>.
 
WLOG let <math>A=(0, 0)</math>, <math>B=(1, 0)</math>, <math>C=(x, y)</math>. Then <math>[ABC]=\frac12|y|</math> and <math>X=(\frac{7x+1}{8}, \frac{7y}{8})</math>, <math>Y=(\frac{x}{8}, \frac{y}{8})</math>, <math>Z=(\frac78, 0)</math>. Then <math>[XYZ]=\frac12|\frac{43y}{64}|</math>. Therefore the answer is <math>\boxed{\frac{43}{64}}</math>.
 
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==Application 2==
and that the solution to this problem by Matholic:
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===Problem 2===
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The Wooga Looga Theorem states that the solution to this problem by Matholic:
  
 
The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF
 
The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF
 
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===Solution 2===
 
is this solution by franzliszt:
 
is this solution by franzliszt:
  

Revision as of 11:51, 29 October 2020

Application 1

Problem 1

The Wooga Looga Theorem states that the solution to this problem by franzliszt:

In $\triangle ABC$ points $X,Y,Z$ are on sides $BC,CA,AB$ such that $\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71$. Find the ratio of $[XYZ]$ to $[ABC]$.

Solution 1

is this solution by RedFireTruck:

WLOG let $A=(0, 0)$, $B=(1, 0)$, $C=(x, y)$. Then $[ABC]=\frac12|y|$ and $X=(\frac{7x+1}{8}, \frac{7y}{8})$, $Y=(\frac{x}{8}, \frac{y}{8})$, $Z=(\frac78, 0)$. Then $[XYZ]=\frac12|\frac{43y}{64}|$. Therefore the answer is $\boxed{\frac{43}{64}}$.

Application 2

Problem 2

The Wooga Looga Theorem states that the solution to this problem by Matholic:

The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF

Solution 2

is this solution by franzliszt:

We apply Barycentric Coordinates w.r.t. $\triangle ABC$. Let $A=(1,0,0),B=(0,1,0),C=(0,0,1)$. Then we find that $D=(\tfrac 56,\tfrac 16,0),E=(0,\tfrac 56,\tfrac 16),F=(\tfrac16,0,\tfrac56)$. In the barycentric coordinate system, the area formula is $[XYZ]=\begin{vmatrix} x_{1} &y_{1}  &z_{1} \\  x_{2} &y_{2}  &z_{2} \\   x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]$ where $\triangle XYZ$ is a random triangle and $\triangle ABC$ is the reference triangle. Using this, we find that\[\frac{[DEF]}{[72]}=\begin{vmatrix} \tfrac 56&\tfrac 16&0\\ 0&\tfrac 56&\tfrac 16\\ \tfrac16&0&\tfrac56 \end{vmatrix}=\frac{7}{12}\] so $[DEF]=42$. $\blacksquare$


The Wooga Looga Theorem can be used to prove many problems and should be a part of any geometry textbook. ~ilp2020

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