1. Similar Polygons - 8. Similarity - Big Ideas Math Geometry, 2014

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

Next, we will mark a point $P$ inside $△ A B C$ 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	$△ A B C$	$△ A^{'} B^{'} C^{'}$	Relation
Angles	$m ∠ A = 4 3^{\circ}$	$m ∠ A^{'} = 4 3^{\circ}$	$m ∠ A = m ∠ A^{'}$
	$m ∠ B = 6 7^{\circ}$	$m ∠ B^{'} = 6 7^{\circ}$	$m ∠ B = m ∠ B^{'}$
	$m ∠ C = 7 0^{\circ}$	$m ∠ C^{'} = 7 0^{\circ}$	$m ∠ C = m ∠ C^{'}$
Sides	$A B = 3.5$	$A^{'} B^{'} = 7$	$\frac{A ^{'} B ^{'}}{A B} = 2$
	$B C = 2.5$	$B^{'} C^{'} = 5$	$\frac{B ^{'} C ^{'}}{B C} = 2$
	$A C = 3.4$	$A^{'} C^{'} = 3.4$	$\frac{A ^{'} C ^{'}}{A C} = 2$
Perimeters	$P_{1} = 9.4$	$P_{2} = 18.8$	$\frac{P _{2}}{P _{1}} = 2$
Areas	$A_{1} = 4.025$	$A_{2} = 16.1$	$\frac{A _{2}}{A _{1}} = 2^{2}$

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

Corresponding angles of $△ A B C$ and $△ A^{'} B^{'} C^{'}$ have the same measure.
Each side length of $△ A^{'} B^{'} C^{'}$ is $twice$ its corresponding side length of $△ A B C .$
The perimeter of $△ A^{'} B^{'} C^{'}$ is $twice$ the perimeter of $△ A B C .$
The area of $△ A^{'} B^{'} C^{'}$ is four times the area of $△ A B C .$

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.

Corresponding angles are congruent.
The ratio of the corresponding side lengths is equal to the scale factor $k .$
The ratio of the perimeters is equal to the scale factor $k .$
The ratio of the areas is equal to the square of the scale factor $k^{2} .$

Exercise 3 Page 417

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