Special Right Triangles

Question

You are hired for a summer job painting the exterior of houses for the elderly in your community. You have completed your on-the-job training and are set to begin your first painting job today. Before you leave, your manager reminds you of the most important rule: Ladders must make an angle of between 40 and 70 degrees with the ground to ensure proper safety.

Upon arriving at the home, you unload your 12- and 18-foot ladders and begin the preparations for painting. Use your information on 45°-45°-90° and 30°-60°-90° right triangles to answer the following questions. Round your answers to the nearest tenth and please show all your work.

1. You wish to use your 12-foot ladder to reach the lower level of the home. How far away from the home must the ladder be placed to form a 45°-45°-90° triangle?
2. For the same 12-foot ladder, how far up on the house will the ladder reach?
3. To reach the second floor, you must use your 18-foot ladder and form a 30°-60°-90° triangle. How far away from the house must the ladder be placed?
4. For the same 18-foot ladder, how far up on the house will the ladder reach?

Answer 1

If you are using the 12 ft ladder to make a 45-45-90 triangle, the 2 legs will be equal, and they must be the hypotenuse divided by sqrt 2. So, the height of the ladder on the house and the distance from the house is 12/sqrt(2) = 8.5 ft

To use the 18 ft ladder to make a 30-60-90 triangle, the ladder must make a 60 degree angle with the ground in order to satisfy the safety regulations. That means the distance on the ground from the house is the short leg. That is half the hypotenuse (18 ft) so that distance is 9 ft. The height on the house is then the long leg which must be short leg (9) times sqrt 3 = 15.6 ft