1. Similar Polygons
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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.
Let's begin by considering a triangle ABC. We will label its side lengths and angle measures, and write its perimeter and area.
Next, we will mark a point P inside △ABC and dilate the triangle by a scale factor of k=2. Let △A′B′C′ be the image of the dilation.
Let's make a table showing relevant information about the triangles.
Parts | △ABC | △A′B′C′ | Relation |
---|---|---|---|
Angles | m∠A=43∘ | m∠A′=43∘ | m∠A=m∠A′ |
m∠B=67∘ | m∠B′=67∘ | m∠B=m∠B′ | |
m∠C=70∘ | m∠C′=70∘ | m∠C=m∠C′ | |
Sides | AB=3.5 | A′B′=7 | ABA′B′=2 |
BC=2.5 | B′C′=5 | BCB′C′=2 | |
AC=3.4 | A′C′=3.4 | ACA′C′=2 | |
Perimeters | P1=9.4 | P2=18.8 | P1P2=2 |
Areas | A1=4.025 | A2=16.1 | A1A2=22 |
From the table above, we can list the following facts.
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 k, we can make four statements.