Determine the length \(x\) of the blue segment
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Solution
In this problem we explore a geometric-algebra challenge: two nested squares and a segment of length \(x\) to determine.
The solution uses triangle similarity and algebraic reasoning to find the value of \(x\).
Follow the clear, step-by-step explanation to understand how geometry and algebra combine to yield the result.
Let’s add sections like it’s shown in the next figure
We have two squares \(\square A B C D \) and \(\square M N G F \)
\[\begin{gathered}
A B \| D C\\\\
and\\\\
E B \| F G\\\\
\angle a=\angle c\\\\
Also\\\\ \triangle B A E \\\\and\\\\ \triangle F C G\\\\ are\; Right\; Triangles.\\\\
\triangle B A E \sim \Delta F C G \\\\
\frac{A B}{C F}=\frac{A E}{C G} \\\\
\frac{7}{4}=\frac{x}{C G} \\\\
C G=\frac{4 x}{7}
\end{gathered}
\]
\[\begin{gathered}
\Delta F C G \;is\; a\; Right \;Triangle\\\\
By\;Pythagoras\;theorem\\\\
F G^{2}=C F^{2}+C G^{2}\\\\
F G=\sqrt{4^{2}+\left(\frac{4 x}{7}\right)^{2}}\\\\
F G=\sqrt{16+\frac{16 x^{2}}{49}}=N G
\end{gathered}
\]
\[\begin{aligned}
&\text { From } \Delta G F C \\\\
&\qquad \begin{array}{c}
\angle a+\angle F C G+\angle C G F=180^{\circ} \\\\
\text { also } \\\\
\qquad \angle b+\angle N G F+\angle C G F=180^{\circ} \\\\
\angle a=\angle b \\\\
\Delta F C G \sim \Delta G N B \\\\
\frac{C F}{N G}=\frac{G F}{B G} \\\\
\frac{4}{N G}=\frac{G F}{B G} \\\\
B G=\frac{1}{4}\left(16+\frac{16 x^{2}}{49}\right)
\end{array}
\end{aligned}
\]
\[\begin{gathered}
B C=B G+G C=7 \\\\
4+\frac{4 x^{2}}{49}+\frac{4 x}{7}=7 \\\\
\frac{4 x^{2}}{49}+\frac{4 x}{7}-3=0 \\\\
4 x^{2}+28 x-147=0 \\\\
x=\frac{-28 \pm \sqrt{28^{2}-4 (4) (-147)}}{2 (4)} \\\\
x=\frac{7}{2}\; \text { or }\; x=-\frac{21}{2} \\\\
\huge x=\frac{7}{2}
\end{gathered}\]
By applying triangle similarity and the Pythagorean theorem, we derived a quadratic equation whose meaningful solution gives the correct value of \(x\).
This problem illustrates how combining geometric insight with algebraic techniques can lead to elegant solutions.
For more visual and algebraic problems like this, explore our Solved Exercises section on Art Of Mathematics.
Home -> Solved problems -> We have two squares, determine the value of x
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