Similarity, Proportion, and Triangle Proofs

Similarity, Proportion, and Triangle Proofs

The Lesson Activities will help you meet these educational goals:

Content Knowledge—You will prove theorems about triangles using similarity relationships.
Mathematical Practices—You will use appropriate tools strategically.
Inquiry—You will perform an investigationin which you will make observations and communicate your results in written form.
STEM—You will apply mathematical and technology tools and knowledge to grow in your understanding of mathematics as a creative human activity.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

Parallel Lines and Similar Triangles

You will use the GeoGebra geometry tool to explore how a line parallel to one side of a triangle divides the other two sides of the triangle. Go to segmentsparallel to sides of a triangle, and complete each step below. If you need help, follow these instructions for using GeoGebra.

Check the box labeled Show Segment Parallel to BC. Notice that intersects two sides of creating a smaller triangle, How is related to ? How do you know?

Sample answer:

Since corresponding angles of parallel lines cut by a transversal are congruent, and . So, based on the AA criterion for similarity, .

What are the lengths of and? Measure and record them.

Sample answer:

AD = 7.5

DB = 18.5

AE = 7.01

EC= 17.28

Find the ratio of ADto DB and the ratio of AEto EC. Round your answers to two decimal places.

Sample answer:

Now select point D, and move it around. You will notice that moves with point D but remains parallel to . Check the box labeled Show Ratios of Intersected Segments to see the ratios : and : . What do you notice about the ratios as you move point D?

Sample answer:

The ratios change as the position of D changes, but the two ratios remain equal to each other.

Now select points B and C, and move them around. What do you notice about as you move B and C?What do you notice about the ratios of the lengths of the intersected segments as you move B and C?

Sample answer:

As points B and C move, moves in such a way that it remains parallel to . The ratiosofthe lengths of the intersected segments change with B and C, but remain equal to each other.

Based on your observations, what conclusion can you draw about the lengths of and?

Sample answer:

The ratio of AD to DB is equal to the ratio of AE to EC. In other words, the pairs of lengths are proportional.

Now check the boxes for Show Segment Parallel to AC and Show Segment Parallel to AB. Find the ratio of BFtoFC and the ratio of GB to AG. Then find the ratio of IC to BI and the ratio of HC to AH. (If you want to move , select point F, and if you want to move , select point H.) What can you say about the relationship between the pairs of lengths in each case?

Sample answer:

The ratio of BFtoFC is proportional to the ratio of GBto AG. Similarly, the ratio of ICtoBI is proportional to the ratio of HCto AH.

Based on your observations in part g, what general conclusion can you draw about any line that is parallel to one side of a triangle and intersects the other two sides?

Sample answer:

A line parallel to one side of a triangle and intersecting the other two sides divides the two sides of the triangle proportionally.

Proportional Sides and Similar Triangles

You will use GeoGebra to determine that a line dividing two sides of a triangle proportionally is parallel to the third side. Go to proportional segments on sidesof a triangle, and complete each step below.

Measure the lengths of and . What is the ratio of AD to DB?

Sample answer:

Measure the length of . Use coordinate algebra to locate a point E on such that the ratio of AE to EC is equal to the ratio of AD to DB. Show how you derived your answer. (Hint: Let x and AC – x represent the two line segments that make up.)

Sample answer:

The length of is 13 units. Let AE = x, and EC =13 – x.

0.6

x = 0.6(13 – x)

x = 7.8 – 0.6x

x + 0.6x = 7.8

1.6x = 7.8

4.875

AE = x= 4.875, and EC = 13 – 4.875 = 8.125.

If I place point E at a distance of 4.875 units from A on —that is, E(5.875,1)—then the ratio of the segments into which E divides is equal to the ratio of AD to DB.

Place point E that you found in part b on . Draw a line passing through D and E, and measure and record the slopes of and .

Sample answer:

Slope of = -0.4

Based on the slopes, what can you say about the relationship between and ?

Sample answer:

and are parallel line segments.

Similar Triangles and the Pythagorean Theorem

By drawing an altitude from the right angle vertex to the hypotenuse of a right triangle, you can divide the triangle into two smaller right triangles. You will use GeoGebra to find the relationship among the three right triangles. Go to proving the Pythagorean Theorem bysimilar triangles, and complete each step below.

What is m in ∆ABC? Record the measurement.

Sample answer:

Place a point D along such that is an altitude of . Using coordinate algebra, find and record the coordinates of point D to satisfy these requirements. Explain how you arrived at your answer.

Sample answer:

D = (9.6, 0). is horizontal with y = 0 at all points on . Also, because is an altitude, it forms a right angle with , so is a vertical line. For all points on , x=9.6.

Check the box labeled Show Altitude of Triangle ABC. The altitude divides into and through the point you determined in part b. Measure and record the side lengths of and . Then measure and record the side lengths of and .

Sample answer:

Side of Triangle / Side of ABC / Length / Side of ADB / Length / Ratio of Sides / Ratio of Sides
hypotenuse / AC / 15 / AB / 12 / / 1.25
short leg / BC / 9 / BD / 7.2 / / 1.25
long leg / AB / 12 / AD / 9.6 / / 1.25
Side of Triangle / Side of ABC / Length / Side of BDC / Length / Ratio of Sides / Ratio of Sides
hypotenuse / AC / 15 / BC / 9 / / 1.667
short leg / BC / 9 / DC / 5.4 / / 1.667
long leg / AB / 12 / BD / 7.2 / / 1.667

What is the relationship between the corresponding sides of and ? What is the relationship between the corresponding sides of and ?

Sample answer:

The corresponding sides of and are proportional. The corresponding sides of and are also proportional.

Because both and are right triangles, and are congruent. Can you identify another pair of congruent angles in the two triangles? Based on these angles, what can you infer about and ? Similarly in and , and are congruent. Can you identify another pair of congruent angles in and ? How can you use your observations to relate and ?

Sample answer:

in and in are congruent because they both have the same measure, so ~based on the AA criterion for similarity. Likewise, in and in are congruent because they both have the same measure. I can also say that ~based on the AA criterion for similarity.

Now select point B, and move it around the screen. You’ll see three checkboxes appear. Check the three boxes to find the ratios of pairs of corresponding side lengths in the two triangles. Note the change in the ratios as you move point B. Fix point B at an arbitrary position, and record the ratios that you see in the table.

Sample answer:

Answers will vary based on the position of point B.

Ratios of Corresponding Side Lengths of Triangles / ABC andADB / ABC and BDC / ADB and BDC
ratio of lengths for hypotenuse / 1.152 / 2.015 / 1.749
ratio of lengths for leg 1 / 1.152 / 2.015 / 1.749
ratio of lengths for leg 2 / 1.152 / 2.015 / 1.749

What can you conclude about and ?

Sample answer:

All three ratios are the same for each pair of right triangles, so~, ~, and ~. Based on this, ~~.