BdMO National Secondary 2007#11
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Find the area of the largest square inscribed in a triangle of sides $5,6$ and $7$.
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Re: BdMO National Secondary 2007#11
There are 3 possibles squares for 3 different bases. Let the base BC be $m$ and side of the square be $x$ .
Now by heron's formula, $[ABC]=6 \sqrt{6}$
Let the vertex of the square B'C'LF in $AB$ be $B'$ and in $AC$ be $C'$ (and F,L in base BC) Now $\angle ABC=90-\angle BB'F =\angle AB'C'$ . Simillarly $\angle ACB= \angle AC'B'$. So $\triangle AB'C' \sim \triangle ABC$
So $\frac{[AB'C']}{[ABC]}=\frac{B'C'^2}{BC^2}=\frac{x^2}{m^2}$. Thus $[ABC]=\frac{6x^2 \sqrt{6}}{m^2}$.
Now, drop a altitude from A to $BC$ and let the feet be $X$. Let the intersection of $B'C'$ and $AX$ be $A'$. As $ B'C' \parallel BC$ and AX is the height of $\triangle ABC$ , AA' is the height of $\triangle AB'C'$ .
Now $\frac{AA'.x}{2}=\frac{6x^2 \sqrt{6}}{m^
2}$
Or,$ AA'= \frac{12x \sqrt{6}}{m^2}$
Simillarly $AX=\frac{12 \sqrt{6}}{m}$.
Now,$AA'+A'X=AX$
Or, $ \frac{12x\sqrt{6}}{m^2}+x=\frac{12 \sqrt{6}}{m}$
Or, $ x= \frac{12m \sqrt{6}}{12\sqrt{6}+m^2}$
Checking m=5,6 and 7 we get m=5 for x being highest and for m=5 , $x^2$ is 7.3 (approx) which is the answer.
Now by heron's formula, $[ABC]=6 \sqrt{6}$
Let the vertex of the square B'C'LF in $AB$ be $B'$ and in $AC$ be $C'$ (and F,L in base BC) Now $\angle ABC=90-\angle BB'F =\angle AB'C'$ . Simillarly $\angle ACB= \angle AC'B'$. So $\triangle AB'C' \sim \triangle ABC$
So $\frac{[AB'C']}{[ABC]}=\frac{B'C'^2}{BC^2}=\frac{x^2}{m^2}$. Thus $[ABC]=\frac{6x^2 \sqrt{6}}{m^2}$.
Now, drop a altitude from A to $BC$ and let the feet be $X$. Let the intersection of $B'C'$ and $AX$ be $A'$. As $ B'C' \parallel BC$ and AX is the height of $\triangle ABC$ , AA' is the height of $\triangle AB'C'$ .
Now $\frac{AA'.x}{2}=\frac{6x^2 \sqrt{6}}{m^
2}$
Or,$ AA'= \frac{12x \sqrt{6}}{m^2}$
Simillarly $AX=\frac{12 \sqrt{6}}{m}$.
Now,$AA'+A'X=AX$
Or, $ \frac{12x\sqrt{6}}{m^2}+x=\frac{12 \sqrt{6}}{m}$
Or, $ x= \frac{12m \sqrt{6}}{12\sqrt{6}+m^2}$
Checking m=5,6 and 7 we get m=5 for x being highest and for m=5 , $x^2$ is 7.3 (approx) which is the answer.