Problem 36. Right Triangles

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Two right triangles \(ABZ\) and \(DBZ\) are given with the edges lengths below:

  • \(\bar{ZA}=33\)

  • \(\bar{BD} =16\)

  • \(\bar{BZ}=65\)

What is the the length of \(\bar{TH}\).

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Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1701844480939397217

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The answer is equal to \(y \approx 11.53613\).
According to the Pythagorean theorem, we have \(\bar{AB}=56\) and \(\bar{DZ}=63\). Considering \(B\) as the origin, we formulate the lines \(\bar{AB}\) and \(\bar{DZ}\) as below. The \(y\) coordinate of the intersection of these two lines is equal to the answer.

\[\begin{align} \left. \begin{aligned} BA:& y = \frac{33}{56} x \\ DZ:& y = -\frac{16}{63}(x - 65) \end{aligned} \right\} \quad &\implies \quad x \approx 19.57647, y \approx 11.53613 \end{align}\]