A cyclist starts from point X and rides 3 km due west to point Y. At Y he changes direction and rides 5km north-west to point Z.

(a) hw far is he from the starting point, correct to the nearest km?
Find the bearing of the Z frm X to the nearest degree?

The angle XYZ = 135o

According to the Cosine Law
ZX=sqr(XY^2+YZ^2-2•XY•YZ•cos135)
=sqr(9+25+30 (-0.707))= 7.43 km ≈7 km.

Yes

To find the distance from the starting point, you can use the Pythagorean theorem because we have a right-angled triangle formed by the cyclist's path.

Let's label the points as follows:
X - starting point
Y - point after riding 3 km due west
Z - final point after riding 5 km northwest

To find the distance from X to Z, we can use the formula:

distance = √(side₁^2 + side₂^2)

In this case, side₁ is the distance between X and Y (3 km) and side₂ is the distance between Y and Z (5 km).

distance = √(3^2 + 5^2)
distance = √(9 + 25)
distance = √34

Rounded to the nearest kilometer, the cyclist is approximately 6 km from the starting point.

Now let's find the bearing of Z from X.

Bearing is measured in degrees clockwise from north. To find the bearing, we can use trigonometry. We need to find the angle formed by the line XZ and the north direction.

tan(angle) = opposite/adjacent

In this case, opposite is side₁ (3 km) and adjacent is side₂ (5 km).

tan(angle) = 3/5
angle = tan^(-1)(3/5)

Using a calculator, angle ≈ 30.96 degrees.

Rounded to the nearest degree, the bearing of Z from X is approximately 31 degrees.

To find the distance from the starting point (X) to the final point (Z), we can use the concept of vector addition.

First, let's represent the displacement towards the west as a vector (-3 km) and the displacement towards the northwest as a vector (-5 km, 45°).

Adding these two displacements, we get the resultant displacement vector from X to Z.

To calculate this vector, we can break it down into its north and west components. The north component can be found by multiplying the magnitude of the vector (5 km) by the sine of the angle (45°). The west component can be found by multiplying the magnitude of the vector (5 km) by the cosine of the angle (45°).

So, the north component is: 5 km * sin(45°) = 5 km * (√2 / 2) = 5√2 / 2 km ≈ 3.5 km
The west component is: 5 km * cos(45°) = 5 km * (√2 / 2) = 5√2 / 2 km ≈ 3.5 km

Since the cyclist traveled 3 km towards the west initially, we subtract 3 km from the west component to get the final west component: 3.5 km - 3 km = 0.5 km

Now we have the north component (3.5 km) and the west component (0.5 km). We can calculate the total displacement using the Pythagorean theorem:

Total displacement = √(north component^2 + west component^2)
= √((3.5 km)^2 + (0.5 km)^2)
= √(12.25 km^2 + 0.25 km^2)
= √(12.5 km^2)
= 3.54 km (rounded to the nearest km)

Therefore, the cyclist is approximately 3.54 km away from the starting point.

To find the bearing of point Z from point X, we can use trigonometry.

Bearing is the angle measured clockwise from the true north direction to the line connecting X and Z. In other words, we need to find the angle formed between the line connecting X and Z and the north direction.

Using the components of the displacement, we can find the angle using the arctan function.

Bearing = arctan(north component / west component)
= arctan(3.5 km / 0.5 km)
≈ 82.85° (rounded to the nearest degree)

Therefore, the bearing of point Z from point X is approximately 83°.

Well, it looks like the cyclist took a bit of a roundabout route, but let's calculate it.

First, we'll use the Pythagorean theorem to find the distance from the starting point:
The cyclist traveled 3 km west and then 5 km northwest, which forms a right-angled triangle.
The horizontal leg of the triangle is 3 km, and the vertical leg is 5 km.

Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the hypotenuse (or the distance from the starting point):
(3^2 + 5^2) = (9 + 25) = 34
Taking the square root of 34 gives us approximately 5.83 km.

So, the cyclist is approximately 5.83 km away from the starting point.

Now, let's find the bearing from X to Z. The bearing is usually measured clockwise from the north direction.

To find the bearing, we need to calculate the angle between the line connecting X and Z and the north direction.

Using trigonometry, we can find this angle:
tan(theta) = opposite/adjacent
tan(theta) = 5/3
theta = atan(5/3)
theta ≈ 59.04 degrees

Therefore, the bearing of Z from X is approximately 59 degrees (to the nearest degree).

Happy cycling!