Similarity, Proof, and Trigonometry

The first set of lessons of this chapter is focused on the study of dilations and similarity transformations. First, the definition of a dilation and its scale factor is presented. Then, using interactive applets, properties of dilations - the dilation of a line is a line, dilations preserve angle measures - are investigated. Once the definition and properties have been set, dilations are applied to different objects like polygons, figures in the coordinate plane, or general images. Students also learn how to perform a dilation using a compass.

Later, similarity transformations are introduced as well as the definition of similar figures. Since similarity transformations are based on dilations, they share the same properties. However, this lesson shows some criteria for determining whether two polygons are similar. Furthermore, triangles come to play, and theorems like the Angle-Angle Similarity Theorem, the Side-Side-Side Similarity Theorem, and the Side-Angle-Side Similarity Theorem are developed. Apart from studying the similarity between two given triangles, some properties within a single triangle are studied and then formalized as theorems - the Triangle Proportionality Theorem and its converse, Right Triangle Proportionality Theorem, Geometric Mean Altitude Theorem, and Geometric Mean Leg Theorem. Additionally, the Pythagorean Theorem is proven using similarity. This set of lessons finishes by investigating different properties of quadrilaterals, such as the fact that the midpoints of the sides of any quadrilateral form a parallelogram. In addition, the Triangle Angle Bisector Theorem and its converse are presented and put into practice.

Next, similarity is applied in right triangles to understand trigonometric ratios. Here, the sine, cosine, and tangent ratios for acute angles are introduced as well as their corresponding inverse and reciprocal ratios. Each of these ratios is put into action to find a missing length in a given triangle. Furthermore, it is shown that the sine and cosine of complementary angles are equal, which helps to rewrite expressions using sine in terms of cosine and vice-versa. Later, the angle of elevation and depression are introduced and used to model different real-world scenarios where trigonometric ratios can be applied. Finally, trigonometric ratios are used to find the volume and density of different objects.

To finish the chapter, three equivalent formulas for finding the area of a triangle using the sine ratio are shown and put into practice. Also, the Law of Sines and the Law of Cosines are stated, proven, and used to find missing measures in a triangle modeling different real-world scenarios.

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