Session 5, Part C:
Steepness and Trigonometry

In This Part: Measuring Steepness | Examining Ratios and Angles | The Tangent

Problem C7

Use a protractor or angle ruler and a ruler to make side-view scale drawings of a ladder leaning against a wall for each of the following situations. Label the measures of and the lengths h and d, and find the height-to-distance ratios. Record your answers in the table below, or print and use the Ladders Worksheet (PDF).

 a. = 45° b. h = 2, d = 1 c. = 30° d. h = 1, d = 2 e. = 60°

Problem

Measure

h:d Ratio

Ratio as Decimal

 a 45° b 2:1 c 30° d 1:2 e 60°

 When given a, start by drawing the angle. Next, choose an integer (a whole number) as the length for the distance from the wall. Draw the right angle and complete the triangle. Finally, measure the height and determine the h:d ratio.   Close Tip When given , start by drawing the angle. Next, choose an integer (a whole number) as the length for the distance from the wall. Draw the right angle and complete the triangle. Finally, measure the height and determine the h:d ratio.

 Problem C8 Examine Problem C7 (b) and (d) above. What do you notice about the h:d ratio and the measure of ?

Problem C9

We've slightly revised the table from Problem C7 to include the 15- and 75-degree angles:

Problem

Measure

Ratio as Decimal

 a 15° 0.27 b 45° 1 c 30° 0.58 d 60° 1.72 e 75° 3.73
 a. Using the data from this table, plot and connect the points on the Steepness Graph of the height-to-distance ratios for a ladder leaning against a wall at different angles. Use the Ladders Worksheet (PDF) to complete the solution. Steepness Graph: b. Examine the information in the Steepness Graph. What happens to as the h:d ratio increases?

 Video Segment In this video segment, participants explore the relationship between the angle of elevation and the height-to-distance ratio. They graph their data to see what happens to the ratio as the angle increases. Were your findings similar? How would you explain this relationship in your own words? If you are using a VCR, you can find this segment on the session video approximately 15 minutes and 43 seconds after the Annenberg Media logo.

 Problem C10 Suppose that it is safe to be on a ladder when the h:d ratio is larger than 2 and smaller than 3. Give a range of angles at which the ladder can be positioned safely.

 Session 5: Index | Notes | Solutions | Video