3
$\begingroup$

We have two squares in a triangle like in the picture. We know that E divides AB into two halves and C divides FG into two halves. Can we somehow determine the ratio of ABJ to BCDE?

enter image description here

I have tried and tried again but there is just one variable that stands between me and the answer. The length of FG.

$\endgroup$
3
  • $\begingroup$ Observe that we know all angles, according to the conditions: $DE/AE=1$ and $BC/CG=3$. $\endgroup$
    – Berci
    Nov 9, 2019 at 11:31
  • $\begingroup$ How do you know the length of CG though? $\endgroup$
    – Pavol Komlos
    Nov 9, 2019 at 11:36
  • $\begingroup$ See my answer. ${{{{}}}}$ $\endgroup$
    – Berci
    Nov 9, 2019 at 11:40

2 Answers 2

Reset to default
1
$\begingroup$

Clearly $\angle BAJ = 45^{\circ}$.

Let $BE=a$ and $FC=b$ and let $v$ be an altitude of $ABJ$ on $AB$.

  • Then $DF = a-b$ and $FI = 2b$ so $a-b=2b\implies a=3b$.
  • Since $ABJ\sim DGJ$ we have ${v-a\over v}= {a+b\over 2a}$ so $v=9b$.

Finnaly we have $${BCDE\over ABJ} = {a^2\over {2a\cdot v\over 2}} = {a\over v} = {3b\over 9b} = {1\over 3}$$

$\endgroup$
1
$\begingroup$

We have $AE=EB=ED$, so the angle $EAD\angle$ is $45^\circ$.
Consequently, we also have $DF=FI$, hence $DF=FG$, thus $F$ is midpoint and $C$ is quarter point of $DG$, yielding $$DG=\frac43DC=\frac23AB\\ FG=\frac13AB$$

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.