Big Ideas Math Geometry, 2014
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Big Ideas Math Geometry, 2014 View details
1. Similar Polygons
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Exercise 3 Page 417

Consider a triangle and perform a dilation. Compare the angles, the side lengths, the areas, and the perimeters of the preimage and the image.

See solution.

Practice makes perfect

Let's begin by considering a triangle We will label its side lengths and angle measures, and write its perimeter and area.

Next, we will mark a point inside and dilate the triangle by a scale factor of Let be the image of the dilation.

Let's make a table showing relevant information about the triangles.

Parts Relation
Angles
Sides
Perimeters
Areas

From the table above, we can list the following facts.

  1. Corresponding angles of and have the same measure.
  2. Each side length of is its corresponding side length of
  3. The perimeter of is the perimeter of
  4. The area of is four times the area of

Note that the relations obtained above do not depend on the measures of the triangle. Therefore, if two triangles are similar and the scale factor is we can make four statements.

  1. Corresponding angles are congruent.
  2. The ratio of the corresponding side lengths is equal to the scale factor
  3. The ratio of the perimeters is equal to the scale factor
  4. The ratio of the areas is equal to the square of the scale factor