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Transcript
CST4_Lesson23_SimTri March 17, 2017 Triangle Geometry Similar Triangles similarity has a lot to do with proportionality the mathematical understanding of SIMILAR is different than our "normal" understanding of the word 1 CST4_Lesson23_SimTri March 17, 2017 Two figures are considered SIMILAR if: ....they have the same shape ....the corresponding angles are congruent ....the corresponding sides are proportional All three of the above conditions must be met When things grow proportionally, we say that they grow in all dimensions by the same factor if something gets three times as wide, it will also get three times as tall and three times as deep 2 CST4_Lesson23_SimTri March 17, 2017 If two parallelograms are proportional, then the corresponding sides are larger (or smaller) by the same factor. scale factor = 5 2 scale factor = 2 5 8.5 cm 3.4 cm 3 cm these are proportional 7.5 cm increasing the size, then k > 1 decreasing the size, then k < 1 A proportion is made of two equal ratios (fractions) from the previous page: 3.4 3 = 8.5 7.5 the measure of a side from "the first" figure the measure of its corresponding side in the second figure 3 CST4_Lesson23_SimTri March 17, 2017 a c = b d a proportion after we cross multiply, these will be equal ad=bc ex 3.4 3 = 8.5 7.5 Two figures are considered SIMILAR if ....they have the same shape this you can verify by looking...is it a triangle, trapezoid, square etc... ....the corresponding angles are congruent this you can verify mathematically ...the measures are either given or can be calculated or deduced ....the corresponding sides are proportional this you can verify mathematically ...set up a proportion 5 8 9 14.4 4 CST4_Lesson23_SimTri March 17, 2017 Remember! Once a triangle has been proven to be similar to another triangle, then you know two things: corresponding angles are congruent corresponding sides are proportional Like proving congruence...we also have minimum conditions to prove SIMILARITY 5 CST4_Lesson23_SimTri March 17, 2017 3 ways to prove that triangles are similar: 1. AA angleangle 2. SSSprop sidesideside (prop) 3. SASprop sideangleside (prop) 1. Proving triangles are similar by AA AngleAngle statement Justification 6 CST4_Lesson23_SimTri March 17, 2017 Now that the 2 triangles are similar by AA ED x k = AB EF x k = BC DF x k = AC where k is the ratio of similarity. sides in one triangle are k times the corresponding sides in the other triangle k can be greater than 1 or less than 1 What are possible values for k ? Example of a question: D Determine the length of segment AB A C 55 94.4 cm 20 cm B 55 E statement 59 cm F justification 7 CST4_Lesson23_SimTri March 17, 2017 Example of a question: D A Determine the length of segment AB mc 02 C B 55 94.4 cm 55 E 59 cm F justification statement 2. Proving triangles are similar by SSSprop sidesideside (prop) 7 10.5 6 9 9 13.5 Statement Justification 8 CST4_Lesson23_SimTri March 17, 2017 Now that the 2 triangles are similar by SSSprop 7 10.5 ED x k = AB EF x k = BC DF x k = AC 6 9 9 13.5 Where k is the ratio of similarity. sides is one triangle are k times the corresponding sides in the other triangle corresponding angles are congruent What are possible values for k ? Example of a question: Prove that C F A D 91 96 174.4 60 E F B C statement justification 9 CST4_Lesson23_SimTri March 17, 2017 3. Proving triangles are similar by SASprop sideangleside (prop) 6 10 8.4 14 justification statement Now that the 2 triangles are similar by SASprop A ED x k = AB EF x k = AC DF x k = BC D 91 96 174.4 60 E F B C Where k is the ratio of similarity. sides in one triangle are k times the corresponding sides in the other triangle What are possible values for k ? 10 CST4_Lesson23_SimTri March 17, 2017 Example of a question: Can you determine the length of EF? A F 14 11 8 D B 5 E C justification statement Example of a question: What is the length of EF? A F 14 11 8 D E 19.25 B 5 C statement justification 11 CST4_Lesson23_SimTri Dealing with OVERLAPPING TRIANGLES March 17, 2017 Many problems involving similar triangles have one triangle ON TOP OF (overlapping) another triangle. Here we have ∆BDE and ∆BAC In this case line segments DE and AC are shown to be parallelnote the arrows on the line segments. so, we know that <BDE is congruent to <DAC (by corresponding angles). <B is shared by both triangles so the two triangles are similar by AA Parallel Line to a Triangle’s Side: Any line parallel to a triangle’s side determines two similar triangles. ∆BXY ∆BAC angles are congruent sides are proportional Corollary: Any two similar triangle determine parallel lines. 12 CST4_Lesson23_SimTri March 17, 2017 Segment Joining the Midpoints of Two Sides in a Triangle: Any segment joining the midpoints of two sides in a triangle is parallel to the third side and is half the measure of this third side. Thales’ Theorem: Two intersecting transversal lines intersected by parallel lines are separated into corresponding segments of proportional length. 13 CST4_Lesson23_SimTri March 17, 2017 A D B E C More OVERLAPPING TRIANGLES A A D D B B C E C E 14 CST4_Lesson23_SimTri March 17, 2017 AB CD EF GH Lines Does not have to be a triangle G E C A O B D F H 15