September 30

Similarity

In Euclidean geometry, the fact that similar triangles exist is equivalent to the Euclidean parallel postulate. In other words, in those geometries where the Euclidean parallel postulate is NOT valid, similar triangles will NOT exist.

Definition of Similar Triangles Two triangles are said to be similar if their corresponding angles are congruent and their corresponding sides are in a constant k proportion. Similarity is an idea only used with geometric figures, and the notation used to indicated similarity between objects is the tilde or "~" sign.

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Similarity Triangle Theorems Theorem 1: If three angles of one triangles are congruent to three angles of another triangle, the triangles are similar. (AAA for similarity) Corollary 1: If two angles of one triangles are congruent to two angles of another triangle, the triangles are similar. (AA for similarity) Corollary 2: If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off a triangle similar to the given triangle. Theorem 2: If two sides of one triangle are in a constant k proportion to two sides of another triangle and the included angle of the first triangle is congruent to the included angle of the second triangle, the two triangles are similar. (SAS for similarity) Theorem 3: If three sides of one triangle are in a constant k proportion to three corresponding sides of another triangle, the triangles are similar. (SSS for similarity)

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Similarity in Sketchpad

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The easiest method to use to work with similarity in Sketchpad is to use the Dilate command under the Transform menu. Steps to construct a similar triangle: 1) Construct any triangle 2) Construct a point outside the triangle and mark it as a center either by double clicking on it or by selecting it and then choosing Mark Center under the Transform menu. 3) Construct two line segments of different lengths. Select the two segments and choose Mark Ratio "m/n" under the Transform menu. NOTE1: The ratio will be marked in the order that you select the two segments. In the above example, since a smaller similar triangle was desired, the smaller segment was chosen first to make the ratio less than 1. If you want to construct a larger similar triangle, you would select the segments in the reverse order. 4) Select your entire triangle and then choose Dilate under the Transform menu. NOTE2: The Dilate window allows you to change your options if you wish. NOTE3: Sometimes, the newly constructed similar object lies on top of the original object. Drag the center point to change the position of the new object. 4) Measure the ration of the segments and then the ration of the corresponding sides. Are they all in the same ration? 5) Measure the corresponding angles. Are the corresponding angles congruent? Steps to subdivide a segment: 1) Construct a segment which you wish to subdivide. 2) Select an endpoint (in the diagram above, point A was selected). Under the Transform menu, select Translate. When the Translate window appears, choose any angle - say 300 - and any distance - say 1 inch. The By Polar Vector will be automatically selected to allow you to choose an angle. 3) A new point, A', will be seen. Translate this point by the same angle and distance to create A". 4) Continue to translate as many points as you need for your subdivision. 5) Construct a segment from your last translated point to the endpoint of your chosen segment. In the diagram above, it would be a segment connecting A''' and B. 6) With the new segment still selected, select your points and construct parallel lines through those points parallel to your end segment. 7) Construct the points of intersection of the parallel lines and your original segment. 8) Measure the distances between the points to determine if those points subdivide the original segment equally.

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