Question

find the tangent of ∠c.
write your answer in simplified, rationalized form. do not round.
tan(c) =

Explanation:

Step1: Recall tangent - ratio definition

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle C$ in right - triangle $CDE$ with right - angle at $D$, the side opposite to $\angle C$ is $DE$ and the side adjacent to $\angle C$ is $CD$. Given $CD = 4$ and $CE=\sqrt{65}$, we first find $DE$ using the Pythagorean theorem.
Let $DE = x$. By the Pythagorean theorem $CD^{2}+DE^{2}=CE^{2}$. So, $4^{2}+x^{2}=(\sqrt{65})^{2}$.

Step2: Solve for the opposite side

We have $16 + x^{2}=65$. Then $x^{2}=65 - 16=49$, and $x = 7$ (we take the positive value since length cannot be negative).

Step3: Calculate the tangent of $\angle C$

Now, $\tan(C)=\frac{DE}{CD}$. Since $DE = 7$ and $CD = 4$, $\tan(C)=\frac{7}{4}$.

Answer:

$\frac{7}{4}$