Question

express tan o as a fraction in simplest terms.

Explanation:

Step1: Find the length of OP

In right triangle \( OPQ \), by the Pythagorean theorem \( OP^{2}+PQ^{2}=OQ^{2} \). Given \( OQ = 50 \), \( PQ=14 \), so \( OP=\sqrt{OQ^{2}-PQ^{2}}=\sqrt{50^{2}-14^{2}}=\sqrt{(50 + 14)(50 - 14)}=\sqrt{64\times36}=\sqrt{64}\times\sqrt{36}=8\times6 = 48 \).

Step2: Calculate \( \tan O \)

The tangent of an angle in a right triangle is \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle O \), opposite side is \( PQ = 14 \), adjacent side is \( OP = 48 \). So \( \tan O=\frac{PQ}{OP}=\frac{14}{48}=\frac{7}{24} \).

Answer:

\(\frac{7}{24}\)