Question

find r.
write your answer as an integer or as a decimal rounded to the nearest tenth.
r =
submit
work it out
not feeling ready yet? these can help:

Explanation:

Step1: Find the third angle of the triangle

The sum of the interior angles of a triangle is \(180^\circ\). We know two angles: \(14^\circ\) and \(144^\circ\). Let the third angle (at \(Q\)) be \(x\). Then:

$$ x = 180^\circ - 14^\circ - 144^\circ = 22^\circ $$

Step2: Apply the Law of Sines

The Law of Sines states that in any triangle, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a, b, c\) are the lengths of the sides opposite angles \(A, B, C\) respectively. In triangle \(PQR\), side \(r\) is opposite the \(14^\circ\) angle, and the side of length \(14\) is opposite the \(22^\circ\) angle. So we have:

$$ \frac{r}{\sin 14^\circ}=\frac{14}{\sin 22^\circ} $$

Step3: Solve for \(r\)

We can rearrange the formula to solve for \(r\):

$$ r=\frac{14\times\sin 14^\circ}{\sin 22^\circ} $$

First, calculate \(\sin 14^\circ\approx0.2419\) and \(\sin 22^\circ\approx0.3746\). Then:

$$ r=\frac{14\times0.2419}{0.3746}=\frac{3.3866}{0.3746}\approx9.0 $$

Answer:

\(9.0\)