The Pythagorean Theorem can also be used to determine lengths of segments that may be formed by multiple right triangles or other polygons. The Pythagorean Theorem will only work to show you the relationship between the side lengths of a right triangle. But once you know that information, you can use it to solve larger problems.

diagram showing boat, lighthouse, and the distances from the problem

The Lighthouse Problem

A boat is sailing parallel to the shore and passes a lighthouse. The beam from the lighthouse is visible for 14 miles. The boat passes the lighthouse along the path shown in the diagram and at its closest point is 11 miles from the lighthouse. For how many miles will the captain of the boat be able to see the lighthouse?

Use the Pythagorean Theorem to set up an equation for k.

Interactive exercise. Assistance may be required.

Now that you have correctly set up your equation, solve for k. Express your answer both as an irrational number and approximated to the nearest tenth of a mile.

Interactive popup. Assistance may be required. Check Your Answer

k = 75miles ≈ 8.7 milesClose Pop Up

Now that we know the value of k, we can use that to answer the original question. For how many miles will the captain of the boat be able to see the lighthouse?

Interactive popup. Assistance may be required.Hint How are the two right triangles in the diagram related?Close Pop Up Interactive popup. Assistance may be required.

Check Your Answer

The captain will be able to see the lighthouse for 17.4 miles.Close Pop Up

diagram showing the outline of the pillow

The Pillow Problem

A large pillow is in the shape of a pentagon as shown. Meredith wants to sew a cord around the edge of the pillow for a decoration. What is the length of the cord that Meredith needs in order to completely cover the edge of the pillow?

coach demonstrating a strategy for her team

Strategy: First, we need to look for right triangles in the picture. Once we have identified the right triangles, we need to determine the dimensions of the right triangles.

Drag the appropriate dimensions to the slots in the figure.

Interactive exercise. Assistance may be required.

Use the figure, along with the Pythagorean Theorem, to calculate the length of each side of the pillow. Enter the values into the text boxes, and then click “submit” to check your answers. If the box is shaded red, then that answer is incorrect. Revise your answer, and click “submit” again.

Interactive exercise. Assistance may be required.

Determine the total length of cord that Meredith should use.

Interactive popup. Assistance may be required.Hint The total length of cord required is equal to the perimeter of the pillow.Close Pop Up

Use the grid below to record your answer. Type your answer in the boxes in front of and behind the decimal. Click inside each box to enter the numeral that belongs in the box, and then click the bubble beneath the numeral to shade the bubble that matches the numeral. After hitting the "Submit" button, incorrect portions of your answer will be shaded gray.

Interactive exercise. Assistance may be required.