Unit 2, Lesson 9
Side Length Quotients in Similar Triangles
Grade 8
9.2
Activity
Triangle is similar to triangles , , and .
The scale factors for the dilations that show triangle is similar to each triangle are in the table.
- Find the side lengths of triangles , , and . Record them in the table.
triangle scale factor length of
short sidelength of
medium sidelength of
long side1 4 5 7 2 3 - Your teacher will assign you 1 of the 3 columns. For all 4 triangles, find the quotient of the triangle side lengths assigned to you and record it in the table.
triangle (long side) (short side) (long side) (medium side) (medium side) (short side) or 1.75 or 1.4 or 1.25 What do you notice about the quotients?
- Compare your results with your partners’ and complete your table.
9.3
Activity
Triangles , , and are all similar.
The side lengths of the triangles all have the same units. Find the unknown side lengths.
Three triangles. Vertices capital A, B C. Sides B C and A, C length 4. Side A, B length lower-case c. Vertices capital I H G. Side I G length lower-case h. Side I H length fraction 12 over 5. Side H G length fraction 6 over 5. Vertices capital D E F. Side E F 5, Side F D length lower-case e, side E D length lowercase d.
Student Lesson Summary
If 2 polygons are similar, then the side lengths in one polygon are multiplied by the same scale factor to give the corresponding side lengths in the other polygon.
For these triangles the scale factor is 2:
Here is a table that shows relationships between the lengths of the short and medium sides of the 2 triangles.
| small triangle | large triangle | |
|---|---|---|
| medium side | 4 | 8 |
| short side | 3 | 6 |
| (medium side) (short side) |
The lengths of the medium side and the short side are in a ratio of . This means that the medium side in each triangle is as long as the short side. This is true for all similar polygons: the ratio between 2 sides in one polygon is the same as the ratio of the corresponding sides in a similar polygon.
We can use these facts to calculate missing lengths in similar polygons. For example, triangles and are similar.
Since side is twice as long as side , side must be twice as long as side . Since is 1.2 units long and , the length of side is 2.4 units.
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