Daily Regents: Right Triangle Altitude Theorem (June 2015)

Even though the title says "Daily", I won't be doing them daily any more, until the exams get closer. But they will be frequent.

Geometry, June 2015, Question 34

34. In the diagram below, the line of sight from the park ranger station, P, to the lifeguard chair, L, on the beach of a lake is perpendicular to the path joining the campground, C, and the first aid station, F. The campground is 0.25 mile from the lifeguard chair. The straight paths from both the campground and first aid station to the park ranger station are perpendicular.

If the path from the park ranger station to the campground is 0.55 mile, determine and state, to the nearest hundredth of a mile, the distance between the park ranger station and the lifeguard chair.

Gerald believes the distance from the first aid station to the campground is at least 1.5 miles. Is Gerald correct? Justify your answer.

First part: PLC is a right triangle. This means that you can find the missing side using the Pythagorean Theorem.
So (PL)² + (LC)² = (LC)²
x² + .25² = .55²
x² + .0625 = .3025
x² = .24
x = 0.48989794855 = 0.49 to the nearest hundredth mile.

The second part of the question wants to know, essentially, what is the length of FC. We already know the length of LC, so we need to find the length of FL. We can do this using the Right Triangle Altitude Theorem.
(PL)² = (FL) * (LC) Remember, we found that x² = .24. Do NOT square 0.49
.24 = FL * .25
.24 / .25 = FL
FL = .96
FC = FL + LC = .96 + .25 = 1.21
Gerald is incorrect. The distance is shorter than 1.5 miles.

Alternate Solution

All three right triangles in the diagram are similar triangles, so their corresponding sides are proportional.
Therefore, the proportion (base1) / (hypontenuse1) = (base2) / (hypontenuse2) must be a true statement.

Fill in the information we know: .25 / .55 = .55 / FC
Cross-multiply: .25 FC = (.55)²
.25 FC = .3025
FC = .3025 / .25 = 1.21, which is less than 1.5