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Transcript
14.1 Ratio & Proportion The student will learn about: similar triangles. 1 1 Triangle Similarity Definition. If the corresponding angles in two triangles are congruent, and the sides are proportional, then the triangles are similar. A D E F B C AB AC BC DE DF EF 2 AAA Similarity Theorem. If the corresponding angles in two triangles are congruent, then the triangles are A similar. D E F B C Since the angles are congruent we need to show the corresponding sides are in proportion. AB AC BC DE DF EF 3 If the corresponding angles in two triangles are congruent, then the triangles are similar. AB AC BC Prove: What will we prove? What given?B=E, C=F Given: is A=D, DE DF EF Construction Why? (1) E’ so that AE’ = DE (2) F’ so that AF’ = DF (3) ∆AE’F’ ∆DEF (4) AE’F =E = B Why? Construction SAS. Why? CPCTE & Given Why? (5) E’F’ ∥ BC (6) AB/AE’ = AC /AF’ Corresponding angles Why? (7) AB/DE = AC /DF Substitute Why? A Why? Prop Thm (8) AC/DF = BC/EF is proven in the same way. QED E’ F’ B C 4 AA Similarity Theorem. If two corresponding angles in two triangles are congruent, then the triangles are similar. A D E F B C In Euclidean geometry if you know two angles you know the third angle. 5 Theorem If a line parallel to one side of a triangle intersects the other two sides, then it cuts off a similar triangle. Don’t confuse this theorem A with If a line intersects two sides of a triangle , and D E cuts off segments proportional to these two sides, then it is parallel to B the third side. Proof for homework. 6 C SAS Similarity Theorem. If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar. A D E F B C 7 If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar. we ~ prove? Given: is AB/DE =AC/DF, A=D What What given? Prove:will ∆ABC ∆DEF Construction Why? (1) AE’ = DE, AF’ = DF (2) ∆AE’F’ ∆DEF (3) AB/AE’ = AC/AF’ (4) E’F’∥ BC (5) B = AE’F’ (6) A = A (7) ∆ABC ∆AE’F’ (8) ∆ABC ∆DEF A QED E’ B Why? SAS Given & substitution (1) Why? Basic Proportion Thm Why? Why? Corresponding angles Reflexive Why? AA Why? Substitute 2 & 7 Why? D F’ E C F 8 SSS Similarity Theorem. If the corresponding sides are proportional, then the triangles are similar. A D E F Proof for homework. B C 9 Right Triangle Similarity Theorem. The altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle. b a c-x A Proof for homework. B 10 Pythagoras Revisited b From the warm up: a c-x B A short side a h x a 2 cx hypotenuse c b a long side b cx h 2 b c (c x) hypotenuse c b a And of course then, a 2 + b 2 = cx + c(c – x) = cx + c 2 – cx = c2 11 Geometric Mean. a b If then biscalledgeometricmeanbetweenaandc. b c It is easy to show that b = √(ac) m CD = 3.99 cm m CE = 5.99 cm E 4 6 6 9 CF = 9.00 cm D C 36 36 F or 6 = √(4 · 9) Construction of the geometric mean. 12 Summary. • We learned about AAA similarity. • We learned about SAS similarity. • We learned about SSS similarity. • We learned about similarity in right triangles. 13 Assignment: 14.1