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5.3 The Ambiguous Case

5.3 The Ambiguous Case. If we are given 2 sides and one angle opposite of one of the sides, we may not get just one answer. There are 3 possibilities … one answer, 2 answers or even no answer! The arrangement of the info is SSA Let’s think about this pictorially before trying out the math.

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5.3 The Ambiguous Case

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  1. 5.3 The Ambiguous Case

  2. If we are given 2 sides and one angle opposite of one of the sides, we may not get just one answer. There are 3 possibilities … one answer, 2 answers or even no answer! The arrangement of the info is SSA Let’s think about this pictorially before trying out the math. For all these examples, we will consider being given m∠A, a, and b & we remember this was not allowed in geometry C a b A B c

  3. C b a A B c ∠A could be acute… and a < b a b a b b a or a A A A 2 solutions if bsinA < a < b 1 solution if a = bsinA No solution if a < bsinA ∠A could be acute… and a ≥ b a b only 1 solution A

  4. C b a A B c ∠A could be obtuse a a b b A A 1 solution if a > b No solution if a ≤ b

  5. Ex 1) Determine the number of solutions. m∠A = 29°, b = 31, a = 23 m∠A is acute m∠A = 132° , b = 96, a = 105 a < b 23 < 31 so 2 solutions bsinA = 31sin29° = 15.02 < 23 < 31 a b m∠A is obtuse and a > b 105 > 96 a b A 1 solution

  6. Ex 2) Solve △ABC if m∠A = 35.18°, c = 17.8 and a = 11.46 B 11.46 17.8 OR ??? 35.18° C A C and here? 180 – 63.49 = 116.51 116.51 + 35.18 = 151.69 < 180 Sure! m∠C = 116.51° Two Answers!

  7. Ex 2) Solve △ABC if m∠A = 35.18°, c = 17.8 and a = 11.46 B 11.46 17.8 35.18° C A m∠C = 63.49° OR m∠C = 116.51° m∠B = 81.33° m∠B = 28.31° b = 19.66 b = 9.43

  8. Ex 3) Solve △ABC if m∠A = 71.4°, a = 45.3 and b = 51.4 C 45.3 51.4 71.4° B A WAIT!! sin has to be between –1 & 1, so… No Triangle exists

  9. Homework #503 Pg 261 #1, 5, 7, 15, 25, 29, 34, 35, 37

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