Find the third angle in triangle PQR.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Find the third angle in triangle PQR.
The sum of angles in a triangle is $180^\circ$.
$$ \angle PQR = 180^\circ - \angle RPQ - \angle QRP $$
$$ \angle PQR = 180^\circ - 60^\circ - 50^\circ $$
$$ \angle PQR = 70^\circ $$
Step 2: Apply the Sine Rule to find PQ.
The Sine Rule states that for a triangle with sides p, q, r and opposite angles P, Q, R:
$$ \frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R} $$
In triangle PQR, we want to find side PQ (let's call it $r$), and we know angle QRP (R). We also know side RP (let's call it $q$) and angle PQR (Q).
$$ \frac{PQ}{\sin(\angle QRP)} = \frac{RP}{\sin(\angle PQR)} $$
Substitute the given values:
$$ \frac{PQ}{\sin(50^\circ)} = \frac{6.5}{\sin(70^\circ)} $$
Solve for PQ:
$$ PQ = \frac{6.5 \times \sin(50^\circ)}{\sin(70^\circ)} $$
$$ PQ = \frac{6.5 \times 0.7660}{0.9397} $$
$$ PQ = \frac{4.979}{0.9397} $$
$$ PQ \approx 5.30 \text{ units} $$
The length of PQ is approximately $\boxed{5.30}$.
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