Question

question
express tan k as a fraction in simplest terms.
answer

Explanation:

Step1: Find the length of side $KL$ using Pythagorean theorem

In right - triangle $KLM$, by the Pythagorean theorem $KL=\sqrt{KM^{2}-LM^{2}}$. Given $KM = 20$ and $LM=18$, then $KL=\sqrt{20^{2}-18^{2}}=\sqrt{(20 + 18)(20 - 18)}=\sqrt{38\times2}=\sqrt{76}=2\sqrt{19}$.

Step2: Recall the definition of the tangent function

The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle K$, the opposite side to $\angle K$ is $LM$ and the adjacent side is $KL$. So $\tan K=\frac{LM}{KL}$.

Step3: Substitute the values of $LM$ and $KL$

Substituting $LM = 18$ and $KL=2\sqrt{19}$, we get $\tan K=\frac{18}{2\sqrt{19}}=\frac{9}{\sqrt{19}}$. Rationalizing the denominator, $\tan K=\frac{9\sqrt{19}}{19}$.

Answer:

$\frac{9\sqrt{19}}{19}$