Question

find the tangent of $\angle d$.
$\sqrt{35}$
$5\sqrt{2}$
write your answer in simplified, rationalized form. do not round.
$\tan(d) = $

Explanation:

Step1: Recall tangent definition

In a right triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\) to \(\theta\). For \(\angle D\), opposite side is \(EC = \sqrt{35}\), adjacent side is \(CD\). First, find \(CD\) using Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c = 5\sqrt{2}\), \(a=\sqrt{35}\), \(b = CD\). So \(CD=\sqrt{(5\sqrt{2})^2 - (\sqrt{35})^2}\).

Step2: Calculate \(CD\)

\[

$$\begin{align*} CD&=\sqrt{50 - 35}\\ &=\sqrt{15} \end{align*}$$

\]

Step3: Find \(\tan(D)\)

\(\tan(D)=\frac{EC}{CD}=\frac{\sqrt{35}}{\sqrt{15}}\). Rationalize the denominator: multiply numerator and denominator by \(\sqrt{15}\).
\[

$$\begin{align*} \tan(D)&=\frac{\sqrt{35}\times\sqrt{15}}{\sqrt{15}\times\sqrt{15}}\\ &=\frac{\sqrt{525}}{15}\\ &=\frac{\sqrt{25\times21}}{15}\\ &=\frac{5\sqrt{21}}{15}\\ &=\frac{\sqrt{21}}{3} \end{align*}$$

\]

Answer:

\(\frac{\sqrt{21}}{3}\)