Question

find the tangent of ∠k.
simplify your answer and write it as a proper fraction, improper fraction, or whole number.
tan(k) =

Explanation:

Step1: Identify the right triangle sides

In right triangle \( \triangle JIK \) with right angle at \( I \), hypotenuse \( JK = 34 \), leg \( IK = 16 \). First, find the other leg \( IJ \) using Pythagorean theorem \( IJ=\sqrt{JK^{2}-IK^{2}} \).
\( IJ=\sqrt{34^{2}-16^{2}}=\sqrt{(34 - 16)(34 + 16)}=\sqrt{18\times50}=\sqrt{900}=30 \)

Step2: Recall tangent definition

Tangent of an angle in a right triangle is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle K \), opposite side is \( IJ = 30 \), adjacent side is \( IK = 16 \).
\( \tan(K)=\frac{IJ}{IK}=\frac{30}{16} \)

Step3: Simplify the fraction

Simplify \( \frac{30}{16} \) by dividing numerator and denominator by their greatest common divisor, which is 2.
\( \frac{30\div2}{16\div2}=\frac{15}{8} \)

Answer:

\( \frac{15}{8} \)