The bearing of two points Q and R from a points P are 030 degree and 130 degree respectively. If PQ is12cm and PR is 5cm, find QR
make that 100° (130-30)
To find the length of QR, we can use the Law of Cosines.
First, let's draw a triangle with points P, Q, and R.
Now, let's label the angles of the triangle:
- The angle at point P is 30 degrees.
- The angle at point Q is 130 degrees.
- The angle at point R is 180 degrees - 30 degrees - 130 degrees = 20 degrees (since the sum of angles in a triangle is 180 degrees).
We are given the lengths of two sides of the triangle:
- PQ = 12 cm
- PR = 5 cm
Using the Law of Cosines, we can find the length of QR:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(angle QPR)
To substitute the values into the equation:
QR^2 = 12^2 + 5^2 - 2 * 12 * 5 * cos(130 - 30)
Simplifying the equation:
QR^2 = 144 + 25 - 120 * cos(100)
QR^2 = 169 - 120 * cos(100)
Now we need to find the cosine of 100 degrees. We can use a scientific calculator or an online calculator to do this. Assuming the cosine of 100 degrees is approximately -0.1736, we can substitute it into the equation:
QR^2 = 169 - 120 * (-0.1736)
QR^2 = 169 + 20.832
QR^2 = 189.832
Finally, we take the square root of both sides to solve for QR:
QR ≈ √189.832
QR ≈ 13.78 cm
Therefore, the length of QR is approximately 13.78 cm.
use the law of cosines:
QR^2 = PQ^2 + PR^2 - 2 PQ PR cos 130