Lesson 4 Cut by a Transversal Solidify Understanding

Jump Start

Each of the following problems is designed to help you identify valid and invalid strategies for changing the form of fractions. Students have suggested that the following equations are true. Examine each problem to determine if the two fractions are equivalent. (You can verify your prediction numerically by calculating decimal equivalents of the fractions on each side of the equation.) If you determine the fractions are not equivalent, what errors in reasoning might have led students to say the fractions are equivalent?

1.

$\frac{2 + 5}{6} = \frac{5}{3}$

2.

$\frac{2 + 5}{6} = \frac{1 + 5}{3}$

3.

$\frac{2 + 5}{6} = \frac{1}{3} + \frac{5}{6}$

Learning Focus

Prove that a line drawn parallel to one side of a triangle that intersects the other two sides divides the other two sides proportionally.

What observations can I make about the segments formed on two sides of a triangle by a line drawn parallel to the third side?

What if the parallel line is below the base?

How might I justify my observations algebraically?

Open Up the Math: Launch, Explore, Discuss

Draw two intersecting transversals on a sheet of lined paper, as in the following diagram. Label the point of intersection of the transversals $A$ . Select any two of the horizontal lines to form the third side of two different triangles. Label the endpoints of the third side of the smaller triangle $B$ and $C$ , and the endpoints of the third side of the larger triangle $D$ and $E$ .

1.

What convinces you that the two triangles formed by the transversals and the horizontal lines are similar? (Note: We can assume that the horizontal lines on a sheet of lined paper are parallel.)

2.

Write some proportionality statements about the sides of the triangles you have drawn, and then verify the proportionality statements by measuring the sides of the triangles.

3.

Select a third horizontal line segment to form a third triangle that is similar to the other two. Write some additional proportionality statements and verify them with measurements.

Tristan has written this proportion for problem 3, based on his diagram below: $\frac{B D}{A B} = \frac{C E}{A C}$

Tia thinks Tristan’s proportion is wrong because some of the segments in his proportion are not sides of a triangle.

4.

Check out Tristan’s idea using measurements of the segments in his diagram.

5.

Now check out this same idea using proportions of segments from your own diagram. Test at least two different proportions, including segments that do not have $A$ as one of their endpoints.

6.

Based on your examples, do you think Tristan or Tia is correct?

7.

Tia still isn’t convinced, since Tristan is basing his work on a single diagram. She decides to start with a proportion she knows is true: $\frac{A D}{A B} = \frac{A E}{A C}$ . (Why is this true?)

Tia realizes that she can rewrite this proportion as $\frac{A B + B D}{A B} = \frac{A C + C E}{A C}$ . (Why is this true?)

Can you use Tia’s proportion to prove algebraically that Tristan is correct?

Ready for More?

Explore how the area of the smaller triangle formed by drawing an interior segment parallel to one of the sides of a triangle is related to the area of the original triangle.

Takeaways

The triangle midline theorem is a special case of a theorem sometimes referred to as “the side-splitter theorem.”

Given: $\overset{―}{U V} ∥ \overset{―}{W X} ∥ \overset{―}{Y Z} ∥ \overset{―}{R T}$

Here are some proportionality statements that can be written based on the side-splitter theorem:

Vocabulary

Lesson Summary

In a previous lesson, we learned that a midline of a triangle, a line that passes through the midpoints of two of the sides, is parallel to the third side and half its length. In this lesson, we extended this theorem to include other segments that cut the sides of a triangle proportionally. We also proved a non-intuitive “side-splitting” theorem about the multiple segments formed when multiple lines parallel to a side of a triangle cut the other two sides of the triangle.

Retrieval

Find all of the missing side lengths in the right triangles. The two triangles are similar to one another.

1.

2.

The line shown has several triangles that can be used to determine the slope. Draw in three slope-defining triangles of different sizes for the line and then create the ratio of rise to run for each of the triangles you draw.

ratios: