Determining Triangle SimilarityDate: 05/26/98 at 22:31:37 From: Big Red Subject: Trigonometry If you have triangle abc with side ab = 65 and side cb = 52, and you have another triangle def with side fe = 40 and side de = 24, how can you tell if they are similar? Date: 05/30/98 at 23:39:35 From: Doctor Mateo Subject: Re: Trigonometry Hi Big Red, The question you ask is about similar triangles. One of the most important things that you need to know about similarity is what it means for two figures to be similar. Similarity means that: (i) the two figures have congruent angles and (ii) the sides are in proportion. To assist you making a determination as to whether two triangles are similar, you might remember from geometry class some theorems about similar triangles. Here are a few important ones: (i) AA~ (angle-angle similarity). This one says that if you know two of the angles of two triangles are congruent, then the third angles of the two triangles are congruent, and the triangles are similar. (ii) SAS~ (side-included angle-side similarity). This one says that if you have two sides that are in proportion (that is, they reduce to the same fraction) and an included angle, then the triangles are similar. (iii) SSS~ (side-side-side similarity). This one says that if the three sides are in proportion (they reduce to the same fraction), then the triangles are similar. At this point you need to ask yourself what you know from the problem or what is presented in a diagram. Let me present several examples to demonstrate the usefulness of the above theorems. (Here m< means "the degree of angle"): Example 1: triangle ABC with m<A = 80 and m<C = 30 and triangle VWX with m<X = 30 and m<V = 80. Similarity exists between the two triangles since m<A = m<V = 80 and m<C = m<X = 30. The third angle of each triangle must measure 70 degrees since the sum of the measures of the interior angles of a triangle sum to 180 degrees. The similarity in this example is then triangle ACB ~ triangle VXW by AA~ theorem. Notice how the vertices match. An 80-degree angle of one triangle matches with an 80-degree angle of another triangle, and a 30-degree angle of one triangle matches with a 30-degree angle of a second triangle. Example 2. triangle MNL with m<N = 90, MN = 25, and NL = 15 and triangle PNL with m<N = 90, PN = 9, and NL = 15. Since the m<N is 90 degrees in both triangles it is probably a good idea to investigate the ratios formed by the segments that create triangles MNL and PNL. A good strategy to employ here might be to write the ratio of the shorter given side of triangle MNL (NL = 15) to the shorter given side of triangle PNL (PN = 9). Then compare it to the ratio of the longer given side of triangle MNL (MN = 25) to the longer given side of triangle PNL (NL = 15). Since the included angle is known to be congruent, then the triangles will be similar if these ratios are equal (which means that the sides are in proportion). So let's see: NL/PN = 15/9 = 5/3 and MN/NL = 25/15 = 5/3 Since both ratios reduce to 5/3 and share an included congruent angle, we have triangle MNL ~ triangle NLP by SAS~. Example 3. triangle ABC with AB = 15, BC = 10, and AC = 20 and triangle FGH with FG = 16, GH = 32, and FH = 24. Since the only information you have is about the length of the sides, we will need to consider the ratio of the lengths of the sides of triangle ABC to the length of the sides of triangle FGH. The best strategy might be to put the lengths of the sides of the triangles in increasing order so that you have: triangle ABC: shortest length ? middle length ? longest length --------------- = ------------- = -------------- triangle FGH: shortest length middle length longest length So in example 3, we have the following: triangle ABC: 10 ? 15 ? 20 ---- = ---- = ---- triangle FGH: 16 24 32 All of the ratios above reduce to 5/8 so the triangles are similar. If you do not get the same reduced fraction for all three ratios of sides then the triangles are not similar. Here triangle ABC ~ triangle HFG by SSS~. Since you have information about two sides you may want to consider what the third side of each of your triangles is. Then use one of these theorems to help you draw a conclusion about your problem. Have fun drawing a conclusion! -Doctor Mateo, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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