Question

use the information given in the figure to find the length $rt$.
the lengths on the figure are not drawn accurately.
in the figure: $qr=105$, $qs=65$, $st=16$, $angle t=90^circ$

Explanation:

Step1: Define variables for lengths

Let $ST=16$, $QS=65$, $QR=105$, $RT=x$, and $QT=y$.

Step2: Apply Pythagoras to $\triangle QTS$

For right triangle $QTS$: $QS^2 = QT^2 + ST^2$
$\Rightarrow 65^2 = y^2 + 16^2$
$\Rightarrow y^2 = 65^2 - 16^2 = 4225 - 256 = 3969$
$\Rightarrow y = \sqrt{3969} = 63$

Step3: Apply Pythagoras to $\triangle QTR$

For right triangle $QTR$: $QR^2 = QT^2 + RT^2$
$\Rightarrow 105^2 = 63^2 + x^2$
$\Rightarrow x^2 = 105^2 - 63^2 = 11025 - 3969 = 7056$
$\Rightarrow x = \sqrt{7056} = 84$

Answer:

84