A cyclist start from a point X and rides 5km due East to a point Y.It changes direction and rides 8km South_East to a point Z.How far is he from the starting point correct to the nearest km.(2)Find the bearing of Z from X to the nearest degree.
Did you make your sketch?
On mine I see the cosine law popping up ....
d^2 = 5^2 + 8^2 - 2(5)(8)cos135·
= ....
d = √.....
for the angle at X,
sinX/8 = sin135°/d
sinX = 8sin135/d
X = ...
How did you get 135°
To find the distance the cyclist is from the starting point, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the cyclist has traveled 5km due East to point Y, and then 8km in a southeasterly direction to point Z. We can consider these two legs of the right-angled triangle formed, where the straight line from X to Z is the hypotenuse.
The distance due East is one leg of the triangle (5km), and the distance traveled in a southeasterly direction is the other leg (8km). We can consider these as the perpendicular and base of the right-angled triangle.
Using the Pythagorean theorem, we can calculate the distance of the hypotenuse (the straight line distance from X to Z) as follows:
Hypotenuse^2 = Perpendicular^2 + Base^2
Hypotenuse^2 = 5^2 + 8^2
Hypotenuse^2 = 25 + 64
Hypotenuse^2 = 89
To find the distance, we need to take the square root of both sides:
Hypotenuse = √89
Hypotenuse ≈ 9.43 km
Therefore, the cyclist is approximately 9.43 km from the starting point.
Now, let's find the bearing of point Z from point X. The bearing represents the direction from the starting point to the destination.
We can use trigonometry to find the angle between the straight line from X to Z and the due East direction (0 degrees).
To find the angle, we need to use the tangent function:
tan(angle) = Base / Perpendicular
In this case, the Base is the distance traveled due East (5km), and the Perpendicular is the distance traveled in a southeasterly direction (8km).
tan(angle) = 5 / 8
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(5 / 8)
angle ≈ 31.80 degrees
Therefore, the bearing of point Z from point X is approximately 31.80 degrees.