In the figure given below, angle pqr equals angle qrs equals angle tur equals 90 degrees. Pq equals 8 sr equals 12 and ur equals 6. Find tu

Question

(https://i.stack.imgur.com/fU02a.jpg)

In the figure given below, angle pqr equals angle qrs equals angle tur equals 90 degrees. Pq equals 8 sr equals 12 and ur equals 6. Find tu

Answer 1

Let's denote $TU= x$ and $QU = y$. Now, from similar triangles, we obtain $$ \left\{ \begin{array}{ccc} \frac{12-x}{6} &= \frac{12}{y+6} &= \frac{x}{y} \\ \frac{8}{y+6} &= \frac{8-x}{y} &= \frac{x}{6} \end{array} \right. \Rightarrow \left\{ \begin{array}{cc} \frac{y}{x} = \frac{y+6}{12} \\ \frac{x}{6} = \frac{8-x}{y} \end{array} \right. \Rightarrow \left\{ \begin{array}{cc} 12y &= xy+ 6x \\ xy &= 48 - 6x \end{array} \right. \Rightarrow 12y = 48 \Rightarrow y = 4 $$ Substituting this to the second line of the first result, we get $$ \frac{8-x}{4} = \frac{x}{6} \Rightarrow x= \frac{24}{5} $$

Answer 2

Hint. Let $QU=x$, $TU=y$. Use similarity of next triangles: $\triangle PQR\sim \triangle TUR$ and $\triangle SRQ\sim \triangle TUQ$. Then: $\frac{8}{6+x}=\frac{y}{6}$ and $\frac{12}{6+x}=\frac{y}{x}$ Comparing them find $x$ and then $y$.