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Worksheet: Applications of Laws of Sines and Cosines

Q1:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where π‘š ∠ 𝐴 = 6 0 ∘ , the perimeter is 156 cm, the length of the small diagonal is 42 cm and 𝐴 𝐡 < 𝐴 𝐷 . Find the area of the ABCD giving the answer to the nearest square centimeter.

  • A 873 cm2
  • B 1 247 cm2
  • C 546 cm2
  • D 624 cm2

Q2:

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.

Q3:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where π‘š ∠ 𝐴 = 7 9 4 2 β€² ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 6 8 4 2 β€² ∘ and 𝐡 𝐷 = 3 2 . 3 c m . Find the perimeter of 𝐴 𝐡 𝐢 𝐷 giving the answer to two decimal places.

Q4:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝐴 𝐡 = 1 0 . 2 c m , π‘š ∠ 𝐢 𝐴 𝐡 = 3 4 1 8 β€² ∘ and π‘š ∠ 𝐷 𝐡 𝐴 = 6 3 1 2 β€² ∘ . Find the length of the diagonal 𝐴 𝐢 giving the answer to two decimal places.

Q5:

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

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

Q6:

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 meter, the distance between the top of the tower and the top of the office building.

  • A 150 m
  • B 105 m
  • C 73 m
  • D 135 m

Q7:

is a trapezoid where , , and . Find giving the answer to the nearest minute.

  • A
  • B
  • C
  • D

Q8:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral where 𝐴 𝐡 = 2 4 c m , 𝐡 𝐢 = 1 8 c m , 𝐢 𝐷 = 9 c m , 𝐴 𝐢 = 3 0 c m , and π‘š ∠ 𝐴 𝐢 𝐷 = 6 8 1 2 β€² ∘ . Find the length of 𝐴 𝐷 to the nearest centimeter and the area of 𝐴 𝐡 𝐢 𝐷 to the nearest square centimeter.

  • 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

Q9:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴 𝐢 = 1 8 c m , π‘š ∠ 𝐴 𝑀 𝐷 = 9 0 6 β€² ∘ and π‘š ∠ 𝐢 𝐴 𝐡 = 3 5 1 2 β€² ∘ . Find the length of 𝐡 𝐷 giving the answer to two decimal places.

Q10:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴 𝐢 = 2 0 . 5 c m , π‘š ∠ 𝐴 𝑀 𝐷 = 1 4 2 ∘ and π‘š ∠ 𝐢 𝐴 𝐡 = 6 5 4 8 β€² ∘ . Find the length of 𝐡 𝐷 giving the answer to two decimal places.

Q11:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where π‘š ∠ 𝐡 = 1 1 4 ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 5 5 ∘ and 𝐡 𝐷 = 2 2 c m . Find the perimeter of 𝐴 𝐡 𝐢 𝐷 giving the answer to two decimal places.

Q12:

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

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

Q13:

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

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