Transcript Lesson 12.1
Section 12-1 How can you find the height of this triangle? One way is to use the fact that corresponding side lengths of similar triangles have the same ratio. For example, in the right triangles shown below, all corresponding angles are congruent, so the triangles are similar. The ratio of the length of the shorter leg to the length of the longer leg is always 0.75, and the ratios of the lengths of other pairs of corresponding sides are also equal. These ratios are called trigonometric ratios. The word trigonometry comes from the Greek words for “triangle” and “measure.” The ratio of the length of the leg opposite the 20.8° to the length of the hypotenuse will be the same in every similar triangle. If you know this ratio, you can solve for the height. Some Old Horse opposite leg sin A hypotenuse Caught Another Horse adjacent leg cos A hypotenuse Taking Oats Away opposite leg tan A adjacent leg Find the unknown length, x. You can write trigonometric ratios using either acute angle of the right triangle. Two hikers leave their campsite. Emily walks east 2.85 km and Savannah walks south 6.03 km. a. After Savannah gets to her destination, she looks directly toward Emily’s destination. What is the measure of the angle between the path Savannah walked and her line of sight to Emily’s destination? b. How far apart are Emily and Savannah? Have you ever noticed that some sets of steps are steeper than others? Building codes and regulations place restrictions on how steep steps can be. Over time these codes change, so stairs built in different locations and at different times may vary quite a bit in their steepness. Refer to the diagram of stairs. According to the 1996 Council of American Building Officials and the 2000 International Code Council, the unit run should be not less than 10 inches, and the unit rise should not be more than 7.75 inches. With these limiting dimensions, what is the angle of inclination for the stairs? A rule of thumb for designing stairs is that the sum of the unit rise and unit run should be about 17.5 inches. Design three different sets of stairs that meet this condition. Make two of your designs within the approved building code given in Step 1. The third design should not meet the building code. Find the angle of inclination for each set of stairs. Consider designing steps to be built alongside Baldwin Street, Dunedin, at an angle of 20.8°. a. How many designs are possible? Do all possible designs meet the code given in Step 1? b. Create a design for the steps that meets the code. Does your design meet the rule of thumb in Step 2? If not, create a new design in which the sum of the rise and run is 17.5 in. Wheelchair ramps are supposed to have a slope between 1/16 and 1/20. For each of these slopes, design a ramp to get up to a door 24 inches above the surrounding ground. What is the angle of each ramp?