The corner A, B, C and D of a ranch are such that B is 8 km directly East of A and C is 6km from B on a bearing of 30 degree.D is 7km From on a bearing of 300 degree.
(a) the bearing of A and D;
(b) the distance BD in kilometers.
(C) the perimeter of the ranch in kilometers.
To find the answers to these questions, we'll use basic trigonometry and geometric concepts. Let's go through each question.
(a) The bearing of A and D:
To find the bearing, we need to determine the angle at which point D is located from point A. The bearing is typically measured in degrees clockwise from North.
First, let's find the direction of D in relation to B:
The bearing of 300 degrees indicates a direction that is 60 degrees clockwise from the North (since 360 - 300 = 60).
Next, let's find the angle between AB and BC:
Since AB is directly East of B, it is at a bearing of 90 degrees. The angle between AB and BC is the difference between 90 degrees and 30 degrees, which is 60 degrees.
Now we can find the angle between AB and AD:
Since AD is in the direction 60 degrees clockwise from North (D's bearing from B), and the angle between AB and BC is 60 degrees, the angle between AB and AD is 120 degrees (60 + 60).
Therefore, the bearing of A and D is 120 degrees.
(b) The distance BD in kilometers:
To find the distance BD, we can use the law of cosines, as we have two sides and the angle between them.
Using the law of cosines, we have:
BD^2 = AB^2 + AD^2 - 2 * AB * AD * cos(120 degrees).
Given that AB = 8 km and AD = 7 km:
BD^2 = 8^2 + 7^2 - 2 * 8 * 7 * cos(120 degrees).
Simplifying:
BD^2 = 64 + 49 - 112 * cos(120 degrees)
= 113.
Taking the square root of both sides:
BD = √113 ≈ 10.63 km.
Therefore, the distance BD is approximately 10.63 km.
(c) The perimeter of the ranch in kilometers:
To find the perimeter of the ranch, we need to sum up the lengths of all the sides.
Given that AB = 8 km, BC = 6 km, and BD = 10.63 km, we can calculate the perimeter:
Perimeter = AB + BC + CD + DA
= 8 km + 6 km + 10.63 km + 7 km
= 31.63 km.
Therefore, the perimeter of the ranch is approximately 31.63 km.
Perimeter 31.8, distance 9.5
Bearing 140