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Worksheet: Applications of Laws of Sines and Cosines

Q1:

is a right triangle at where and . Find the length of and the angles and giving the answer to the nearest degree.

  • A , ,
  • B , ,
  • C , ,
  • D , ,
  • E , ,

Q2:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴 𝐢 = 2 1 . 1 c m , π‘š ∠ 𝐴 𝑀 𝐷 = 8 0 5 4 β€² ∘ and π‘š ∠ 𝐢 𝐴 𝐡 = 5 3 5 4 β€² ∘ . Find the area of the parallelogram giving the answer to two decimal places.

Q3:

𝐴 𝐡 𝐢 is a triangle, where 𝐷 is the midpoint of 𝐡 𝐢 , π‘š ∠ 𝐡 = 2 8 ∘ , π‘š ∠ 𝐴 = 6 9 ∘ , and π‘Ž = 2 0 c m . Find the length of 𝐴 𝐷 giving the answer to two decimal places.

Q4:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral where π‘š ∠ 𝐴 𝐡 𝐢 = 9 0 ∘ , π‘š ∠ 𝐡 𝐴 𝐷 = 4 1 ∘ , 𝐴 𝐡 = 𝐴 𝐷 = 3 0 . 9 c m and 𝐡 𝐷 = 𝐡 𝐢 . Find the area of 𝐴 𝐡 𝐢 𝐷 giving the answer to two decimal places.

Q5:

is a right triangle at where and . Find the length of to two decimal places and then the measure of angles and to the nearest second.

  • A , ,
  • B , ,
  • C , ,
  • D , ,

Q6:

In a parallelogram 𝐴 𝐡 𝐢 𝐷 , 𝐡 𝐢 = 8 i n c h e s and 𝐴 𝐡 = 5 i n c h e s , where π‘š ∠ 𝐴 𝐡 𝐢 = 1 3 4 ∘ . Work out the length of 𝐴 𝐢 . Give your answer to 2 decimal places.

Q7:

Find the area of 𝐴 𝐡 𝐢 𝐷 given 𝐸 is the point of intersection of 𝐴 𝐢 and 𝐡 𝐷 , 𝐴 𝐸 = 5 c m , 𝐸 𝐢 = 8 . 9 c m , 𝐸 𝐷 = 7 . 7 c m , and π‘š ∠ 𝐴 𝐸 𝐡 = 8 0 ∘ . Give the answer to the nearest square centimeter.

  • A 369 cm2
  • B 185 cm2
  • C 40 cm2
  • D 92 cm2
  • E 19 cm2

Q8:

is a right triangle at where and . Find the lengths of and giving the answer to two decimal places and the measure of giving the answer to the nearest degree.

  • A , ,
  • B , ,
  • C , ,
  • D , ,

Q9:

is a right triangle at where and . Find the lengths and giving the answer to two decimal places and the measure of angle giving the answer to the nearest degree.

  • A , ,
  • B , ,
  • C , ,
  • D , ,

Q10:

is a right-angled triangle at where and . Find in radians and lengths and giving all answers to three decimal places.

  • A , ,
  • B , ,
  • C , ,
  • D , ,

Q11:

The side length of a regular pentagon 𝐴 𝐡 𝐢 𝐷 𝐸 is 25.81 cm. Find the length of the diagonal 𝐴 𝐢 giving the answer to two decimal places.

Q12:

𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐴 = 3 0 ∘ , the ratio between 𝑏 and 𝑐 is √ 3 ∢ 2 and the area of the circumcircle is 2 2 5 πœ‹ cm2. Find the perimeter of the triangle 𝐴 𝐡 𝐢 giving the answer to the nearest centimeter.

  • A 36 cm
  • B 142 cm
  • C 24 cm
  • D 71 cm

Q13:

is a trapezium, where , , , , and . Find the lengths of and giving the answer to the nearest centimetre.

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q14:

𝐴 𝐡 𝐢 is a triangle where 𝐴 𝐡 = 7 c m , π‘š ∠ 𝐡 = 6 0 ∘ and the area of the triangle is 9 1 √ 3 cm2. Find the perimeter of 𝐴 𝐡 𝐢 giving the answer to two decimal places.

Q15:

is a right triangle at where and . Find in radians and the lengths and giving all answers to three decimal places.

  • A , ,
  • B , ,
  • C , ,
  • D , ,

Q16:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 2 6 cm, 𝑏 = 2 2 cm and 𝑐 = 6 cm. Find the radius of the circumcircle giving the answer to three decimal places.

Q17:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 4 4 c m , 𝑏 = 3 1 c m and π‘š ∠ 𝐢 = 6 9 ∘ . Point 𝐷 lies on 𝐴 𝐡 such that 𝐢 𝐷 βŸ‚ 𝐴 𝐷 . Find the length of 𝐢 𝐷 giving the answer to two decimal places.

Q18:

is a trapezium where , , , and . Find the area of the trapezium giving the answer to the nearest square centimetre.

Q19:

is a triangle, where cm, cm, and . Find the radius of the circumcircle giving the answer to two decimal places.

  • A 20.01 cm
  • B 15.60 cm
  • C 29.71 cm
  • D 10.00 cm

Q20:

is a trapezoid, where , , , , and . Find the area of the trapezoid giving the answer to the nearest square centimeter.

  • A 106 cm2
  • B 853 cm2
  • C 355 cm2
  • D 427 cm2
  • E
    1 706
    cm2

Q21:

𝐴 𝐡 𝐢 is a triangle where s i n s i n s i n 𝐴 2 5 = 𝐡 2 9 = 𝐢 1 5 . Find the largest angle in 𝐴 𝐡 𝐢 giving the answer to the nearest degree.

Q22:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral where 𝐴 𝐡 = 1 6 c m , π‘š ∠ 𝐴 𝐷 𝐡 = 4 0 ∘ , π‘š ∠ 𝐷 𝐡 𝐴 = 1 0 0 ∘ , 𝐡 𝐢 = 2 1 c m and 𝐷 𝐢 = 9 c m . Find π‘š ∠ 𝐡 𝐢 𝐷 giving the answer to the nearest second and the area of 𝐡 𝐢 𝐷 giving the answer to three decimal places.

  • A 1 1 1 1 0 β€² 6 β€² β€² ∘ , 88.123 cm2
  • B 2 3 3 3 β€² 2 3 β€² β€² ∘ , 37.767 cm2
  • C 6 9 2 3 β€² 5 8 β€² β€² ∘ , 134.283 cm2
  • D 4 5 1 6 β€² 3 0 β€² β€² ∘ , 67.142 cm2

Q23:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral where 𝐴 𝐡 = 1 4 c m , π‘š ∠ 𝐴 𝐷 𝐡 = 7 0 ∘ , π‘š ∠ 𝐷 𝐡 𝐴 = 4 0 ∘ , 𝐡 𝐢 = 2 6 c m and 𝐷 𝐢 = 2 7 c m . Find π‘š ∠ 𝐡 𝐢 𝐷 giving the answer to the nearest second and the area of 𝐡 𝐢 𝐷 giving the answer to three decimal places.

  • A 7 0 4 6 β€² 1 1 β€² β€² ∘ , 331.415 cm2
  • B 7 8 4 0 β€² 1 9 β€² β€² ∘ , 344.162 cm2
  • C 6 4 2 9 β€² 5 0 β€² β€² ∘ , 356.909 cm2
  • D 3 0 3 3 β€² 3 0 β€² β€² ∘ , 178.454 cm2

Q24:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral where 𝐴 𝐡 = 7 c m , π‘š ∠ 𝐴 𝐷 𝐡 = 6 1 ∘ , π‘š ∠ 𝐷 𝐡 𝐴 = 5 8 ∘ , 𝐡 𝐢 = 2 4 c m and 𝐷 𝐢 = 1 9 c m . Find π‘š ∠ 𝐡 𝐢 𝐷 giving the answer to the nearest second and the area of 𝐡 𝐢 𝐷 giving the answer to three decimal places.

  • A 1 2 8 3 6 β€² 4 8 β€² β€² ∘ , 178.154 cm2
  • B 3 8 1 2 β€² 4 8 β€² β€² ∘ , 141.038 cm2
  • C 6 0 5 2 β€² 0 β€² β€² ∘ , 103.923 cm2
  • D 1 3 1 0 β€² 2 5 β€² β€² ∘ , 51.963 cm2

Q25:

𝐴 𝐡 𝐢 is an isosceles triangle where 𝐴 𝐡 = 𝐴 𝐢 = 4 7 c m and 𝐡 𝐢 = 1 0 c m . Find the angles in the triangle giving the answer to the nearest second.

  • A π‘š ∠ 𝐴 = 1 6 7 5 1 β€² 1 8 β€² β€² , π‘š ∠ 𝐡 = 6 4 β€² 2 1 β€² β€² , π‘š ∠ 𝐢 = 6 4 β€² 2 1 β€² β€² ∘ ∘ ∘
  • B π‘š ∠ 𝐴 = 1 6 7 4 7 β€² 1 0 β€² β€² , π‘š ∠ 𝐡 = 6 6 β€² 2 5 β€² β€² , π‘š ∠ 𝐢 = 6 6 β€² 2 5 β€² β€² ∘ ∘ ∘
  • C π‘š ∠ 𝐴 = 1 2 8 β€² 4 2 β€² β€² , π‘š ∠ 𝐡 = 8 3 5 5 β€² 3 9 β€² β€² , π‘š ∠ 𝐢 = 8 3 5 5 β€² 3 9 β€² β€² ∘ ∘ ∘
  • D π‘š ∠ 𝐴 = 1 2 1 2 β€² 5 0 β€² β€² , π‘š ∠ 𝐡 = 8 3 5 3 β€² 3 5 β€² β€² , π‘š ∠ 𝐢 = 8 3 5 3 β€² 3 5 β€² β€² ∘ ∘ ∘