Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of geometry wikipedia , lookup
Penrose tiling wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Apollonian network wikipedia , lookup
Technical drawing wikipedia , lookup
Rational trigonometry wikipedia , lookup
Multilateration wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
Euclidean geometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Warm up FIND THE UNKNOWN VALUES: Lesson 37: Similar Triangles 40o 25 4 y xo 10 x zo yo 100o Similar is a mathematical word meaning the same shape. We say that two triangles, ΔFDE and ΔLMK, are similar if the ratios of corresponding sides are equal. This equation of three term ratios can also be written in fraction form: FD DE EF LM = MK = KL Corresponding sides are equal FD : DE : EF = LM : MK : KL NOTE: The sides of one triangle must be in the numerator, while the sides of the other triangle are in the denominator. ΔFDE≈ΔLMK Prove the two triangles are similar. In similar triangles, corresponding angles are equal. ∠F = ∠ L ∠D = ∠ M ∠E = ∠ K IMPORTANT: If you know two triangles are similar, then their corresponding angles are equal. Conversely, if two triangles have equal corresponding angles, then the triangles are similar. SOLUTION: Sides: Angles: BC AB ∠ A = ∠ D DE = EF = ∠ B = ∠ E 8 ∠ C = ∠ F 12 = 69 = AC DF 5 7.5 Since corresponding angles are equal, then the triangles are similar. We could also say that since corresponding sides are equal, the triangles are similar. 1 Find the measures of ∠ A, ∠ B, ∠ C. ΔABC is similar to ΔRPQ. Find the lengths of RP and AC. SOLUTION: Sides: FG = GH FH = BC AB AC 3.5 3 = 7 6 = 5.3 10.6 1 2 = 1 2 1 2 = SOLUTION: Sides: 8 x 12x = 8(16.8) x = Therefore: ∠ F = ∠ A ∠ H = ∠ C ∠ G = ∠ B 134.4 12 RP = 11.2 y 12 = 21 16.8 12 = y 21 16.8 16.8y = 12(21) 252 y = 16.8 AC = 15 o ΔSIP is similar to ΔMAT. Find the lengths of MA and SP = 8 = 12 16.8 x Since the ratios of the sides are all equal, the triangles are similar. Thus, the corresponding angles are equal. ∠ A = 100 o ∠ C = 38 o ∠ B = 42 AC AB BC = = RQ RP PQ For the figure below, show that ΔDEF is similar to ΔDGH. Remember: If two triangles have equal corresponding angles, then the triangles are similar. SOLUTION: Reason ∠ D = ∠ D ∠ E = ∠ G ∠ F = ∠ H common angle corresponding ∠s corresponding ∠s ΔDEF ≈ ΔDGH. 2