1. Module 6

1.13. Page 3

Mathematics 10-3 Module 6 Lesson 3

Module 6: Triangles and Other Polygons

 

Explore

 

In the previous lessons you explored a variety of methods for drawing similar polygons. Now you will explore three additional methods for drawing similar triangles.

 

Try This

 

Work with a partner, if possible.

 

Method 1: Constructing Congruent Angles

 

Step 1: Draw any

 

This illustration is of a triangle labelled ABC.

 

Step 2: Use your ruler to draw the base of This base can be any length. If you want the sides of to be twice as long as those of you would make But any length will do.

 

Note: Using the names (instead of completely different letters) to label the new triangle makes it easy to see the pairs of corresponding angles that are the same size. In addition, this way of labelling makes it clear which sides correspond to one another.

 

Step 3: Use your protractor to draw angles at congruent to Call the point where these angles’ arms cross point Join

 

This illustration shows triangle ABC at the left, and the construction of triangle A'B'C' at the right. There are protractors at B' and C' showing these angles are congruent to angles B and C.

 

Now you will check whether the two triangles are similar.

 

TT 4. You used your protractor to make the corresponding angles congruent. That is, you made Now listen to “How to Read a Symbolic Geometric Statement.”

 

Why must Check with your protractor to see if Record those measures on your diagram.

 

TT 5. Now you will check to see if the ratios of the corresponding sides are equal. Measure, and then record on your diagram, the lengths, to the nearest millimetre, of the sides of both triangles.

 

Calculate and compare Are the triangles similar?

 

Method 2: Constructing Proportional Sides

 

Step 1: Draw any

 

This illustration is of a triangle labelled ABC.

 

Step 2: Decide what ratio you wish to use. If you want a similar triangle with sides twice as long, begin by measuring the base BC. Then draw base twice as long as BC.

 

This illustration shows triangle ABC with BC horizontal. To the right of triangle ABC is line segment B'C', which is twice as long as side BC of triangle ABC.

 

Step 3: Measure AB. Then set your compass to a radius twice as large. With eqn081.eps as centre, draw an arc where is likely to be.

 

This illustration shows triangle ABC with BC horizontal. To the right of triangle ABC is line segment B'C', which is twice as long as side BC of triangle ABC. An arc is shown above B’C’ with centre B'. The radius of the arc is twice the length of BA.

 

Step 4: Measure AC. Then set your compass to a radius twice as large. With as centre, draw an arc intersecting the first arc. Call this point Join

 

This illustration shows triangle ABC to the left of triangle A'B'C'. It shows an arc drawn with centre B’ and radius twice the length of side BA. It also shows an arc drawn with centre C' and radius twice the length of side CA. A' is placed where the two arcs meet.

 

TT 6. You decided to make the sides twice as long, but you could have used any ratio—three times, one-half, or any value of your choosing. You know the sides are proportional, but are the corresponding angles congruent? Measure the angle pairs. Record their measures on your diagram. Are the triangles similar? Explain your answer.

 

Method 3: Constructing One Pair of Congruent Angles and Two Pairs of Proportional Sides Forming That Angle

 

This method is often used if the two triangles share an angle.

 

Step 1: Draw any triangle ABC.

 

This illustration shows a triangle labelled ABC.

 

Suppose you want the second triangle to share Also suppose that you want the sides of the second triangle to be one-third as long as the sides of

 

Step 2: Measure AB. Divide that length by 3. Measure out that distance from point A along AB. Call that point So,

 

Step 3: Measure AC. Divide that length by 3. Measure out that distance from point A along AC. Call that point So,

 

Step 4: Join

 

This illustration shows triangle ABC with a segment B'C' drawn inside triangle ABC. B' is one third of the distance from A to B and C' is one third of the distance from A to C.

 

 

 

 

 

But are similar? To find out, answer these questions.

 

TT 7.

 

TT 8.

 

TT 9.

 

TT 10. Are and BC parallel? Justify your answer.

 

TT 11. Which method of drawing similar triangles makes the most sense to you? Why?

 

Share

 

It is time to share your answers to TT 4 to TT 11. Remember that sharing work is an important part of learning. To make the most of your sharing opportunity, be sure that you do the following:

  • Check that you have completed TT 4 to TT 11 to the best of your ability and have the answers in a form that you can easily share with another student, or with your teacher, if so directed.

  • Use the discussion area for your class, or another method indicated by your teacher, to post your answers to TT 4 to TT 11 and to view the work of the people with whom you are sharing.

  • Compare answers for TT 4 to TT 11. Identify where you have similar answers and where your answers are different. Discuss all differences between your answers until you have an agreement. If necessary, you may wish to involve your teacher in your discussion.

  • Revise your answers to TT 4 to TT 11 where necessary.

Save the revised copy of your work, including your diagrams, in your course folder. Ask your teacher whether you are to save a transcript of your discussion in your course folder as well.