Final answer:
In triangle ΔKPS, if m∠P = 105°, m∠S = 30°, and PS = 12, the length of PK is approximately 12.84.
Step-by-step explanation:
In triangle ΔKPS, the sum of the angles is always 180°.
So, to find the measure of angle K, we can subtract the sum of angles P and S from 180°. Angle K = 180° - m∠P - m∠S = 180° - 105° - 30° = 45°.
Now, we can use the Law of Sines to find the length of PK.
The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is the same for all sides of a triangle.
In triangle ΔKPS, we can set up the proportion:
PS/sin(m∠K) = PK/sin(m∠P)
Substituting the values we know:
12/sin(45°) = PK/sin(105°)
Solving for PK:
PK = 12 * sin(105°)/sin(45°)
Using a calculator, we find that PK ≈ 12.84.
So therefore the value of PK is 12.84.