Question

Ocean waves move in parallel lines toward the shore. The figure shows the path that a windsurfer takes across several waves. For this exercise, think of the windsurfer's wake as a line. If m∠1 = (8x + 5y)° and m∠2 = (8x + y)°, find x and y.

78°

Answer 1

x = 9.75 and y = 0

Explanation:

Since the path of the windsurfer crosses both angles, we can use the fact that the sum of the angles around a point is 180 degrees to set up an equation:

m∠1 + m∠2 + m∠3 = 180°

We know that m∠1 = (8x + 5y)° and m∠2 = (8x + y)°. We can also see from the diagram that m∠3 = 180° - 78° = 102°. Substituting these values into the equation, we get:

(8x + 5y)° + (8x + y)° + 102° = 180°

Simplifying and solving for x and y, we get:

16x + 6y = 78°

8x = 78° - 6y

x = (78° - 6y)/8

We can substitute this expression for x into one of the earlier equations to solve for y:

8[(78° - 6y)/8] + y = 78°

78° - 6y + y = 78°

-5y = 0

y = 0

Now that we know y = 0, we can substitute this value into the equation we found for x:

x = (78° - 6(0))/8 = 9.75