Final answer:
To find the length of PK in triangle KPS with angles P and S known, calculate angle K using the sum of angles in a triangle. Then use trigonometry, specifically the sine function and the known side PS, to solve for PK which is approximately 16.97 units.
Step-by-step explanation:
To solve for the length of side PK in triangle △KPS, we can use the Law of Sines or standard trigonometric principles. However, since we have not been provided with the length of side KS or angle K, the Law of Sines cannot be immediately applied. Instead, we can use the fact that the angles of a triangle sum up to 180 degrees to find the measure of angle K.
First, calculate the measure of angle K:
m∠K = 180° - (m∠P + m∠S) = 180° - (105° + 30°) = 45°.
Now, use trigonometry to solve for side PK. Since we know angle S and its opposite side PS, we can use the sine function:
PK = PS / sin(∠K) = 12 / sin(45°).
Remember to use a calculator set to degree mode to find the sine of 45 degrees, which is approximately 0.7071. Multiplying that by the length of PS gives the length of PK:
PK ≈ 12 / 0.7071 ≈ 16.97 units.
Therefore, the length of side PK is approximately 16.97 units.