Worksheet: Applications of Laws of Sines and Cosines

In this worksheet, we will practice deciding whether the rule of sines or the rule of cosines is most appropriate for solving a non-right triangle problem.

Q1:

𝑀 is a circle with radius 24 cm. A chord is drawn whose central angle is 6 2 . Find the length of the chord giving the answer to the nearest cm.

Q2:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝐴 𝐵 = 1 0 . 2 c m , 𝑚 𝐶 𝐴 𝐵 = 3 4 1 8 and 𝑚 𝐷 𝐵 𝐴 = 6 3 1 2 . Find the length of the diagonal 𝐴 𝐶 giving the answer to two decimal places.

Q3:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝐴 𝐵 = 4 . 3 c m and the diagonals 𝐴 𝐶 and 𝐵 𝐷 make angles of 4 9 and 9 4 respectively with the side 𝐴 𝐵 . Find the length of the diagonals giving the answer to three decimal places.

  • A 𝐵 𝐷 = 8 . 1 6 0 c m , 𝐴 𝐶 = 1 4 . 2 5 5 c m
  • B 𝐵 𝐷 = 5 . 3 9 2 c m , 𝐴 𝐶 = 7 . 1 2 8 c m
  • C 𝐵 𝐷 = 1 0 . 7 8 5 c m , 𝐴 𝐶 = 1 8 . 8 4 2 c m
  • D 𝐵 𝐷 = 1 0 . 7 8 5 c m , 𝐴 𝐶 = 1 4 . 2 5 5 c m

Q4:

𝐴 𝐵 𝐶 𝐷 is a trapezium where 𝐴 𝐷 𝐶 𝐵 , 𝐴 𝐷 = 4 c m , 𝐴 𝐵 = 1 7 c m and 𝑚 𝐵 𝐴 𝐷 = 1 0 8 . Find 𝑚 𝐷 𝐵 𝐶 giving the answer to the nearest minute.

  • A 1 1 4 7
  • B 2 9 4 7
  • C 7 1 2 2
  • D 6 0 1 3

Q5:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑚 𝐴 = 6 0 , the perimeter is 156 cm, the length of the small diagonal is 42 cm and 𝐴 𝐵 < 𝐴 𝐷 . Find the area of the ABCD giving the answer to the nearest square centimetre.

Q6:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴 𝐶 = 1 8 c m , 𝑚 𝐴 𝑀 𝐷 = 9 0 6 and 𝑚 𝐶 𝐴 𝐵 = 3 5 1 2 . Find the length of 𝐵 𝐷 giving the answer to two decimal places.

Q7:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑚 𝐴 = 7 9 4 2 , 𝑚 𝐷 𝐵 𝐶 = 6 8 4 2 and 𝐵 𝐷 = 3 2 . 3 c m . Find the perimeter of 𝐴 𝐵 𝐶 𝐷 giving the answer to two decimal places.

Q8:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑚 𝐵 = 1 1 4 , 𝑚 𝐷 𝐵 𝐶 = 5 5 and 𝐵 𝐷 = 2 2 c m . Find the perimeter of 𝐴 𝐵 𝐶 𝐷 giving the answer to two decimal places.

Q9:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝐴 𝐵 = 1 4 c m , 𝐵 𝐶 = 2 8 c m , 𝑚 𝐴 𝐵 𝐶 = 9 0 , 𝑚 𝐵 𝐶 𝐷 = 6 9 and 𝑚 𝐶 𝐷 𝐴 = 8 4 . Find the lengths of 𝐴 𝐷 and 𝐶 𝐷 giving the answer to two decimal places.

  • A 𝐴 𝐷 = 2 8 . 0 0 c m , 𝐶 𝐷 = 2 5 . 3 2 c m
  • B 𝐴 𝐷 = 2 5 . 3 2 c m , 𝐶 𝐷 = 2 1 . 2 4 c m
  • C 𝐴 𝐷 = 2 1 . 2 4 c m , 𝐶 𝐷 = 1 4 . 0 0 c m
  • D 𝐴 𝐷 = 2 1 . 2 4 c m , 𝐶 𝐷 = 2 5 . 3 2 c m

Q10:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝐴 𝐵 = 2 4 c m , 𝐵 𝐶 = 1 8 c m , 𝐶 𝐷 = 9 c m , 𝐴 𝐶 = 3 0 c m , and 𝑚 𝐴 𝐶 𝐷 = 6 8 1 2 . Find the length of 𝐴 𝐷 to the nearest centimetre and the area of 𝐴 𝐵 𝐶 𝐷 to the nearest square centimetre.

  • A 𝐴 𝐷 = 2 8 c m , 467 cm2
  • B 𝐴 𝐷 = 2 8 c m , 266 cm2
  • C 𝐴 𝐷 = 2 2 c m , 485 cm2
  • D 𝐴 𝐷 = 2 8 c m , 341 cm2

Q11:

The side length of a regular octagon is 39.7 cm. Find the length of the diagonals 𝐻 𝐵 , 𝐻 𝐶 , and 𝐻 𝐷 , giving your answer to three decimal places.

  • A 𝐻 𝐵 = 1 0 3 . 7 4 1 c m , 𝐻 𝐶 = 9 5 . 8 4 4 c m , 𝐻 𝐷 = 7 3 . 3 5 6 c m
  • B 𝐻 𝐵 = 7 3 . 3 5 6 c m , 𝐻 𝐶 = 1 0 5 . 2 4 1 c m , 𝐻 𝐷 = 1 1 2 . 4 8 c m
  • C 𝐻 𝐵 = 6 8 . 7 6 2 c m , 𝐻 𝐶 = 7 9 . 4 c m , 𝐻 𝐷 = 8 8 . 7 7 2 c m
  • D 𝐻 𝐵 = 7 3 . 3 5 6 c m , 𝐻 𝐶 = 9 5 . 8 4 4 c m , 𝐻 𝐷 = 1 0 3 . 7 4 1 c m

Q12:

The height of a tower is 139 m and the height of an office building is 54 m. From a point on level ground between them, the angle of elevation of the top of the tower is 6 8 and the angle of elevation of the top of the office building is 4 8 . Find, to the nearest metre, the distance between the top of the tower and the top of the office building.

Q13:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴 𝐶 = 2 8 . 7 c m , 𝑚 𝐴 𝑀 𝐷 = 9 2 2 4 and 𝑚 𝐶 𝐴 𝐵 = 6 3 5 4 . Find the length of 𝐵 𝐷 giving the answer to two decimal places.

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