Lesson 2

Discover

 

You may not have a totem pole outside your home; however, if a nearby power pole or light standard casts a shadow along a level stretch of ground on a sunny day, you can determine its height by using similar right triangles.

 

Try This 1

 

You will need a partner to help you and a measuring tape that has both SI and imperial units.

 

Step 1: On a sunny day, measure the length of the shadow cast by a power pole or light standard. At the same time, measure the length of the shadow cast by a vertical object of known height. The object could be a fence post, a hockey stick, or even your partner.

 

Step 2: Draw a diagram and record your measures. A sample diagram is shown below.

 

1. On your diagram, describe what objects you chose and the date and time of day when you took your measurements.
   
   
2. Explain why the Sun’s rays shown in the diagram must strike the ground at the same angle in each triangle.
   
   
3. Explain how you know the two right triangles are similar.
   
   
4. Set up a proportion and solve for the height of the pole.

Save your responses to your course folder.

 

Share 1

 

Share your responses to the questions in Try This 1 with a classmate or with a group of people.

 

How are your responses to the questions similar? How do your answers differ?

 

If required, save a copy of your discussion in your course folder.

 

Explore

 

In Discover, the two right triangles you used were similar because one pair of acute angles were equal in measure.

 

Suppose you had two right triangles, , and in those triangles . Then, .

 

 

The reason is based on the principle that in a right triangle, the two acute angles must add up to 90°.

 

For ,
Because angle B is equal to 90°,   the two acute angles, A and C, must also equal 90°.
For example, if ∠C = 40°, then ∠A must be 50°.
Since the two triangles are similar, the corresponding angles will be the same.	∠F = 40° and ∠D = 50°

 

Try This 2

 

Jonathan walked 15 m from the wall of his apartment building. At that point, he noticed that his shadow reached the same point on the ground as the building’s shadow. If Jon is 1.6 m tall, and his shadow is 3 m long, how high is the apartment building? Round your answer to one decimal place.

1. Sketch and label a diagram to represent the problem.

Watch the steps involved in solving this problem in the Apartment Shadow Problem. On the final slide of the animation, you will set up the proportion needed to solve this problem.

2. Solve the proportion to find out how high the apartment building is.

Save your responses in your course folder.

 

Example

 

A roof truss is built with a slope of 2.5:12. What is the vertical height of the truss in the diagram? Round your answer to one decimal place.

 

 

Solution

 

The two right triangles representing the slope and the truss are similar because the truss must have the same slope. Corresponding sides will have the same ratios and, therefore, all the angles will be equal.   Let x be the height of the truss.   Set up a proportion for the corresponding sides of the triangles.
Solve the proportion.   The height is approximately 2.1 ft or 2 ft 1 in.

 

Self-Check 1

 

1. A wheelchair ramp is 30 m long, and its bottom is 1.5 m below the top of the ramp.
   
   
   1. How far is a wheelchair above the foot of the ramp when it is 10 m from the top of the ramp? Answer
      
      
   2. What is the slope of the ramp? Round your answer to two decimal places. Answer

2. Vanaja is standing on the east bank of the South Saskatchewan River in Saskatoon. She looks at the historic Bessborough Hotel on the opposite shore and wonders how far away the building is. Vanaja walks 150 m downstream along the bank and stops and places a stick, S, in the bank. Then she walks 30 m further downstream. She turns 90° and walks 20 m away from the river until she sees that the stick she placed on the bank lines up with the hotel across the river. Calculate the original distance, x, that Vanaja was from the hotel.
   
   
   
   Answer
   
   

3. A timber is positioned 30 cm from the end of a 2-m steel bar. The free end of the bar is lifted 20 cm. To the nearest centimetre, how high is the timber off the ground?
   
   
   
   Answer
   
   
4. The foot of a 20-ft ladder is located 5 ft from a vertical wall. A painter is standing 16 ft above the foot of the ladder. How far from the wall is he?
   
   
   iStockphoto/Thinkstock
   
   Answer

Self-Check 2

 

Connect

 

Going Beyond

 

This puzzle involves deductive reasoning and some knowledge of similar triangles and slope.

 

Print two copies of this Missing Square Puzzle. The two triangles appear to be identical—the base of each triangle is 13 units and the height of each one is 5 units. To convince yourself that each triangle is made up of identical pieces, cut out the pieces from the upper triangle in one copy that you printed. Then fit the pieces from the top triangle onto the bottom triangle in your other copy.

 

Where does the missing square come from?

 

Lesson 2 Assignment

 

Lesson 2 Summary

 

How many similar triangles can you see in this drawing? How many are right triangles?

 

In this lesson you saw many similar right triangles. You reviewed their properties and found that if two right triangles had one acute angle with the same measurement, then the right triangles were similar. You also used similar right triangles to solve a variety of problems.