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Ambiguous Case Triangles

Ambiguous Case Triangles. Can given numbers make a triangle ? Can a different triangle be formed with the same information?. Conditions for Unique Triangles. AAS. ASA. two angles must sum to less than 180 º. two angles must sum to less than 180 º. SSS. SAS.

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Ambiguous Case Triangles

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  1. Ambiguous Case Triangles Can given numbers make a triangle? Can a different triangle be formed with the same information?

  2. Conditions for Unique Triangles AAS ASA two angles must sum to less than 180º two angles must sum to less than 180º SSS SAS two shortest sides are longer than the third side Any set of data that fits these conditions will result in one unique triangle.

  3. This diagram is deceiving -- side-side-angle data may result in two different triangles. b a SSA A Ambiguous Triangle Case(aka the ‘bad’ word) Side a is given but it might be possible to ‘swing’ it to either of two positions depending on the other given values. An acute or an obtuse triangle may be possible.

  4. Solve for sin B Set up Law of Sines Find mB in quadrant I Find mB in quadrant II mB  46º mB  180 – 46 = 134º Find mC Find mC mC  (180 – 17 – 134)  29º mC  (180 – 17 – 46)  117º Example (2 triangles) Giveninformation mA = 17º a = 5.8 b = 14.3 Both values of C are possible, so 2 triangles are possible

  5. Solve for sin B Set up Law of Sines Find mB in quadrant I Find mB in quadrant II mB  25º mB  180 – 25 = 155º Find mC Find mC mC  (180 – 58 – 25)  97º mC  (180 – 58 – 155)  -33º Example (1 triangle) Given information mA = 58º a = 20 b = 10 Only one value of C is possible, so only 1 triangle is possible

  6. Solve for sin B Set up Law of Sines Example (0 triangles) Given information mA = 71º a = 12 b = 17 No value of B is possible, so no triangles are possible

  7. Law of Sines Method 1) Use Law of Sines to find angle B • -If there is no value of B (for example, sin B = 2), then there are notriangles Remember, sin x is positive in both quadrant I and II 2) Determine value of B in quadrant II (i.e. 180 – quadrant I value) 3) Figure out the missing angle C for both values of angle B by subtracting angles A and B from 180 4) If it is possible to find angle C for -both values of B, then there are 2 triangles • -only the quadrant I value of B, then only 1 triangle is possible

  8. Solve a triangle where a = 5, b = 8 and c = 9 Draw a picture. This isSSS 9  5 Do we know an angle and side opposite it? No, so we must use Law of Cosines. 84.3   8 Let's use largest side to find largest angle first. minus 2 times the productof those other sides times the cosine of the angle between those sides One side squared sum of each of the other sides squared CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

  9. How can we find one of the remaining angles? Do we know an angle and side opposite it?  9 62.2 5 84.3   33.5 8 Yes, so use Law of Sines.

  10. OR….. • If you have SSA: (h = b Sin A) • No triangles exist if: • a < h • a < b -- One triangle exists if: • a = h • a > b • Two triangles exist if: • h < a < b

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