hep geometry

Find the length UV.

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The length of the diagram $\overline{QR}$  is a.

$\overline{ST}=2\cdot \overline{TR}+5\\ \overline{ST}+\overline{TR}=30\\ 2\cdot \overline{TR}+5+\overline{TR}=30\\ \overline{TR}=\frac{25}{3}$

$f(x)= -\frac{30-\frac{25}{3}}{a}\cdot x+30 =-\frac{65}{3a}\cdot x+30\\ g(x)= \frac{30}{a}\cdot x \\ f(x)=g(x)\\-\frac{65}{3a}\cdot x+30 = \frac{90}{3a}\cdot x\\ \frac{155}{3a}\cdot x=30\\ x=\frac{90a}{155}=\frac{18a}{31}$

$g(\frac{18a}{31})=\overline{UV}=\frac{30}{a}\cdot\frac{18a}{31}=17.42$

$The\ length\ \overline{UV}\ is\ 17.42$

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