Question

find the tangent of ∠p.

right triangle pqr with right angle at r, rq = 48, pq = 52

simplify your answer and write it as a proper fraction, improper fraction, or whole number.

tan(p) =

Explanation:

Step1: Find the length of PR

In right triangle \( PRQ \), by the Pythagorean theorem \( PQ^2 = PR^2 + RQ^2 \). We know \( PQ = 52 \) and \( RQ = 48 \). So,
\( PR^2 = PQ^2 - RQ^2 = 52^2 - 48^2 \)
\( PR^2 = (52 - 48)(52 + 48) \) (using difference of squares: \( a^2 - b^2=(a - b)(a + b) \))
\( PR^2 = 4\times100 = 400 \)
\( PR=\sqrt{400}=20 \)

Step2: Calculate \( \tan(P) \)

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. For \( \angle P \), the opposite side is \( RQ = 48 \) and the adjacent side is \( PR = 20 \). So,
\( \tan(P)=\frac{\text{opposite}}{\text{adjacent}}=\frac{RQ}{PR}=\frac{48}{20} \)
Simplify \( \frac{48}{20} \) by dividing numerator and denominator by 4: \( \frac{48\div4}{20\div4}=\frac{12}{5} \)

Answer:

\( \frac{12}{5} \)