1) A jogger runs directly east for 5 km, then turns and goes northwest for 7 km. He then travels directly south for 5 km. How far is he from the starting point? (km)

2) In what direction is he from the starting point(measured as an angle counterwise from the east axis, units are deg)? (Northwest is the direction lying exactly half way between north and west.)

To determine the distance the jogger is from the starting point, we can break down the jogger's movements into their respective components:

1) Distance travelled east: 5 km
2) Distance travelled northwest: 7 km

To find the jogger's total distance from the starting point, we can use the Pythagorean theorem. The formula is:

Distance = √(Distance east^2 + Distance northwest^2)

Plugging in the values we have:
Distance = √(5^2 + 7^2)
Distance = √(25 + 49)
Distance = √74
Distance ≈ 8.60 km

Therefore, the jogger is approximately 8.60 km away from the starting point.

To determine the direction the jogger is from the starting point, we can use trigonometry. We can calculate the angle θ using the tangent function:

θ = tan^(-1)(opposite/adjacent) = tan^(-1)(Distance northwest/Distance east)

Plugging in the values we have:
θ = tan^(-1)(7/5)
θ ≈ 53.13 degrees

Therefore, the jogger is approximately 53.13 degrees counterwise from the east axis, which is in the northeast direction.

To find the distance from the starting point, we can visualize the jogger's path and use the Pythagorean theorem.

Step 1: Draw a diagram representing the jogger's path.

Let's assume the starting point is at the origin (0,0) on a coordinate plane. The jogger runs directly east for 5 km, so he ends up at point A (5,0). Then, he turns and goes northwest for 7 km, which means he travels in the direction between north and west. Let's assume this direction is 45 degrees. So, he ends up at point B (5 + 7cos45, 7sin45) = (8.5, 4.95). Finally, he travels directly south for 5 km, which means he ends up at point C (8.5, 4.95 - 5) = (8.5, -0.05).

Step 2: Apply the Pythagorean theorem.

The distance between the starting point (0,0) and the final point C (8.5, -0.05) is given by the formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the coordinates, we get:
Distance = √((8.5 - 0)^2 + (-0.05 - 0)^2)
Distance = √(72.25 + 0.0025)
Distance = √72.2525
Distance = 8.505 km

Therefore, the jogger is approximately 8.505 km from the starting point.

Now, let's find the direction in degrees.

Step 1: Determine the angle between the jogger's final position and the positive x-axis.

To calculate this angle, we can use the inverse tangent function. The angle can be calculated as:

Angle = atan((y2 - y1) / (x2 - x1))

Substituting the coordinates, we get:

Angle = atan((-0.05 - 0) / (8.5 - 0))
Angle = atan(-0.05 / 8.5)
Angle ≈ -0.003 radians

Step 2: Convert the angle to degrees.

To convert the angle from radians to degrees, multiply it by 180/π (where π is approximately 3.14159):

Angle_deg = -0.003 * (180/π)
Angle_deg ≈ -0.172 degrees

Therefore, the jogger is approximately 0.172 degrees counter-clockwise from the east axis, which can be rounded to -0.2 degrees.

1. Northwest = (90+180)/2 = 135 deg.

X = hor = 5km + 7cos135 = 0.05km.

Y = ver = -5km + 7sin135 = -0.05km.

d = sqrt((0.05)^2 + (0.05))^2 = 0.71km.
= distance from starting point.

2. tanAr = Y/X = -0.05 / 0.05 = -1,
Ar = -77.58 deg. = Reference angle.
A = 180 + (-77.58) = 102.4 deg. = Direction.