Lesson Worksheet: Ambiguous Case of the Law of Sines Mathematics

In this worksheet, we will practice using the rule of sines to solve SSA (side-side-angle) ambiguous triangles.

Q1:

𝐴𝐵𝐶 is a triangle, where 𝑚𝐵=110, 𝑏=16 cm, and 𝑐=12 cm. How many possible solutions are there for the other lengths and angles?

  • Ano solutions
  • Bone solution
  • Ctwo solutions

Q2:

𝐴𝐵𝐶 is a triangle where 𝑎=13.8cm, 𝑏=15.9cm and 𝑚𝐴=28. Find all possible values for the other lengths and angles giving lengths to two decimal places and angles to the nearest second.

  • A𝑐=51.29cm, 𝑚𝐵=324445, 𝑚𝐶=1191515 or 𝑐=4.86cm, 𝑚𝐵=1471515, 𝑚𝐶=44445
  • B𝑐=25.65cm, 𝑚𝐵=1191515, 𝑚𝐶=324445 or 𝑐=2.43cm, 𝑚𝐵=44445, 𝑚𝐶=1471515
  • C𝑐=25.65cm, 𝑚𝐵=324445, 𝑚𝐶=1191515 or 𝑐=2.43cm, 𝑚𝐵=44445, 𝑚𝐶=1471515
  • D𝑐=25.65cm, 𝑚𝐵=324445, 𝑚𝐶=1191515 or 𝑐=2.43cm, 𝑚𝐵=1471515, 𝑚𝐶=44445

Q3:

𝐴𝐵𝐶 is a triangle, where 𝑚𝐴=40, 𝑎=17cm, and 𝑏=23cm. If the triangle exists, find all the possible values for the other lengths and angles giving the lengths to two decimal places and angles to the nearest second.

  • A𝑐=11.11cm, 𝑚𝐵=793454, 𝑚𝐶=60256
  • B𝑐=26.01cm, 𝑚𝐵=60256, 𝑚𝐶=793454
  • C𝑐=26.01cm, 𝑚𝐵=60256, 𝑚𝐶=793454 or 𝑐=9.23cm, 𝑚𝐵=1193454, 𝑚𝐶=20256
  • D𝑐=11.11cm, 𝑚𝐵=793454, 𝑚𝐶=60256 or 𝑐=14.25cm, 𝑚𝐵=1193454, 𝑚𝐶=20256

Q4:

𝐴𝐵𝐶 is a triangle, where 𝑚𝐴=55, 𝐵𝐶=13cm, and 𝐴𝐶=28cm. If the triangle exists, find all the possible values for the other lengths and angles in 𝐴𝐵𝐶, giving the lengths to two decimal places and angles to the nearest degree.

  • A𝐴𝐵=22.94 cm, 𝑚𝐵=90, and 𝑚𝐶=35
  • B𝐴𝐵=24.80 cm, 𝑚𝐵=90, and 𝑚𝐶=35
  • CThe triangle does not exist.

Q5:

For the given figure, 𝐴𝐵=11, 𝐵𝐶=9, and 𝑚𝐵𝐴𝐶=41. Use the law of sines to work out the measure of 𝐴𝐶𝐵. Give your answer to two decimal places.

Q6:

𝐴𝐵𝐶 is a triangle, where 𝑚𝐴=70, 𝐵𝐶=3cm, and 𝐴𝐶=39cm. If the triangle exists, find all the possible values for the other lengths and angles in 𝐴𝐵𝐶 giving the lengths to two decimal places and angles to the nearest degree.

  • A𝐴𝐵=𝐴𝐵=38.88cm, 𝑚𝐵=90, 𝑚𝐶=20
  • BThe triangle does not exist.
  • C𝐴𝐵=𝐴𝐵=36.65cm, 𝑚𝐵=90, 𝑚𝐶=20

Q7:

𝐴𝐵𝐶 is a triangle where 𝑚𝐴=40, 𝑎=5cm and 𝑏=4cm. If the triangle exists, find all the possible values for the other lengths and angles in 𝐴𝐵𝐶 giving lengths to two decimal places and angles to the nearest second.

  • A𝑐=7.35cm, 𝑚𝐵=305646, 𝑚𝐶=109314
  • B𝑐=3.40cm, 𝑚𝐵=305646, 𝑚𝐶=109314 or 𝑐=2.77cm, 𝑚𝐵=149314, 𝑚𝐶=9314
  • C𝑐=3.40cm, 𝑚𝐵=305646, 𝑚𝐶=109314
  • D𝑐=7.35cm, 𝑚𝐵=109314, 𝑚𝐶=305646
  • ENo triangle exists.

Q8:

𝐴𝐵𝐶 is a triangle, where 𝑚𝐵=70, 𝑏=3 cm, and 𝑐=6 cm. How many possible solutions are there for the other lengths and angles?

  • Aone solution
  • Bno solutions
  • Ctwo solutions

Q9:

𝐴𝐵𝐶 is a triangle, where 𝑚𝐵=130, 𝑏=17 cm, and 𝑐=3 cm. How many possible solutions are there for the other lengths and angles?

  • Ano solutions
  • Bone solution
  • Ctwo solutions

Q10:

Given an isosceles triangle 𝐴𝐵𝐶 with sides 𝐴𝐵=𝐴𝐶, the side 𝐵𝐶=2, and the 𝐴=80, find the length of the side 𝐴𝐵 approximate to the nearest one decimal place.

Q11:

𝐴𝐵𝐶 is a triangle with sides 𝐴𝐵=𝑐, 𝐴𝐶=𝑏, and 𝐵𝐶=𝑎. If the perimeter of triangle 𝐴𝐵𝐶=𝑝, which of the given relations can we use to write the perimeter in terms of the sine of angles using the sine rule?

  • A𝑝=𝑎𝐴+𝐵+𝐶𝐴sinsinsinsin
  • B𝑝=𝑎𝐴𝐵𝐶𝐴sinsinsinsin
  • C𝑝=𝑏𝐴+𝐵+𝐶𝐴sinsinsinsin
  • D𝑝=𝐴+𝐵+𝐶𝐴sinsinsinsin
  • E𝑝=𝑐𝐴+𝐵+𝐶𝐴sinsinsinsin

Q12:

In a triangle 𝐴𝐵𝐶, 𝑎=3cm, 𝑏=5cm, and 𝑚𝐴=120. How many triangles can be formed?

  • ATwo triangles
  • BOne triangle
  • CZero triangles
  • DThree triangles
  • EAn infinite number of triangles

Q13:

For a triangle 𝐴𝐵𝐶, 𝑎=5cm, 𝑏=9cm, and 𝑚𝐴=25. How many triangles can be formed?

  • AAn infinite number of triangles
  • BTwo triangles
  • CThree triangles
  • DOne triangle
  • EZero triangles

Q14:

In a triangle 𝐴𝐵𝐶, 𝑎=2cm, 𝑏=4cm, and 𝑚𝐴=30. How many triangles can be formed?

  • AAn infinite number of triangles
  • BThree triangles
  • CTwo triangles
  • DOne triangle
  • EZero triangles

Q15:

In a triangle 𝐴𝐵𝐶, 𝑎=5cm, 𝑏=5cm, and 𝑚𝐴=95. How many triangles can be formed?

  • AThree triangles
  • BOne triangle
  • CAn infinite number of triangles
  • DTwo triangles
  • EZero triangles

Q16:

In the given figure, if 𝐴 is an acute angle and 𝑎=, how many triangles can be formed?

  • ATwo triangles
  • BOne triangle
  • CThree triangles
  • DZero triangles
  • EAn infinite number of triangles

Q17:

In the given figure, if 𝐴 is an obtuse angle and 𝑎>𝑏, how many triangles can be formed?

  • AAn infinite number of triangles
  • BTwo triangles
  • COne triangle
  • DZero triangles
  • EThree triangles

Q18:

For a triangle 𝐴𝐵𝐶, 𝑎=2cm, 𝑏=5cm, and 𝑚𝐴=35. How many triangles can be formed?

  • AThree triangles
  • BNo triangles can be formed
  • CTwo triangles
  • DOne triangle
  • EAn infinite number of triangles

Q19:

For a triangle 𝐴𝐵𝐶, 𝑎=5cm, 𝑏=4cm, and 𝑚𝐴=90. How many possible solutions are there for each of the other lengths and angles?

  • AOne solution
  • BZero solutions
  • CTwo solutions
  • DThree solutions
  • EAn infinite number of solutions

Q20:

For a triangle 𝐴𝐵𝐶, 𝑎=6cm, 𝑏=5cm, and 𝑚𝐴=40. How many triangles can be formed?

  • AAn infinite number of triangles
  • BThree triangles
  • COne triangle
  • DTwo triangles
  • ENo triangles can be formed

Q21:

For a triangle 𝐴𝐵𝐶, 𝑎=2cm, 𝑏=5cm, and 𝑚𝐵=40. Find all the possible measures of 𝐴 to the nearest degree.

Q22:

For a triangle 𝐴𝐵𝐶, 𝑎=3cm, 𝑏=9cm, and 𝑚𝐴=10. Find all the possible measures of 𝐵 to the nearest degree.

  • A149 and 25
  • B31
  • C31 and 149
  • D31 and 25
  • E25

Q23:

For a triangle 𝐴𝐵𝐶, 𝑎=5cm, 𝑏=5cm, and 𝑚𝐴=80. How many possible solutions are there for each of the other lengths and angles?

  • AOne solution
  • BAn infinite number of solutions
  • CThree solutions
  • DTwo solutions
  • EZero solution

Q24:

𝐴𝐵𝐶 is a triangle in which 𝑎=5cm, 𝑏=8cm, and 𝑚𝐴=36. If the triangle exists, find all the possible values of angle 𝐵 to the nearest second.

  • AThe triangle does not exist.
  • B𝑚𝐵=70742
  • C𝑚𝐵=70742 or 𝑚𝐵=1095118
  • D𝑚𝐵=70718 or 𝑚𝐵=1095142
  • E𝑚𝐵=1095118

Q25:

For the triangle 𝐴𝐵𝐶, 𝑎=3cm, 𝑏=2cm, and 𝑚𝐴=95. Find all the possible measures of 𝐵 to the nearest degree.

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