To determine the requested angles, we can use trigonometry and geometry principles:
1. Acute angle between QR and the x-axis:
The angle between QR and the x-axis can be found by calculating the arctan of the slope of QR. The slope (m) can be found using the formula: m = (y2 - y1) / (x2 - x1).
Given points Q(4, 2) and R(7, -5):
m(QR) = (-5 - 2) / (7 - 4) = -7/3
Using the arctan function, we can find the acute angle (A1):
A1 = arctan(-7/3) ≈ -68.2 degrees
2. Angle of inclination of PQ:
The angle of inclination of PQ can be found using the same method as above with points P(-1, -1) and Q(4, 2):
m(PQ) = (2 - (-1)) / (4 - (-1)) = 3/5
The angle of inclination (A2) can be found by taking the arctan of the slope:
A2 = arctan(3/5) ≈ 30.96 degrees
3. Angle of inclination of POR:
The angle of inclination of POR can be found using points P(-1, -1) and R(7, -5):
m(POR) = (-5 - (-1)) / (7 - (-1)) = -6/8 = -3/4 = -0.75
The angle of inclination (A3) can be found using the arctan function:
A3 = arctan(-0.75) ≈ -36.87 degrees
4. Angle of inclination of SP:
To find the angle of inclination of SP, we can use points S(-3, -7) and P(-1, -1):
m(SP) = (-1 - (-7)) / (-1 - (-3)) = 6/2 = 3
The angle of inclination (A4) can be found using the arctan function:
A4 = arctan(3) ≈ 71.57 degrees
5. Angle SPR:
To find angle SPR, we can use the Law of Cosines. Using the distance formula, we can find the lengths of sides SP, SR, and PR. Let's denote SP as a, SR as b, and PR as c:
a = sqrt((-3 - (-1))^2 + (-7 - (-1))^2) = sqrt(4^2 + 6^2) = sqrt(52) ≈ 7.21
b = sqrt((-3 - 7)^2 + (-7 - (-5))^2) = sqrt(10^2 + 2^2) = sqrt(104) ≈ 10.2
c = sqrt((-1 - 7)^2 + (-1 - (-5))^2) = sqrt((-8)^2 + 4^2) = sqrt(80) ≈ 8.94
Now we can apply the Law of Cosines to find the angle SPR (A5):
cos(A5) = (a^2 + b^2 - c^2) / (2ab)
A5 = arccos((7.21^2 + 10.2^2 - 8.94^2) / (2 * 7.21 * 10.2)) ≈ 42.1 degrees
6. Angle PSR:
To find angle PSR, we can subtract the angle SPR (A5) from the angle of inclination of SP (A