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Lesson: Applications of Laws of Sines and Cosines

Worksheet • 25 Questions

Q1:

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.

Q2:

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.

Q3:

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.

Q4:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where 𝐴 𝐡 = 4 4 c m and 𝐡 𝐢 = 6 8 c m . Find the length of 𝐴 𝐢 to two decimal places and then the measure of angles 𝐴 and 𝐢 to the nearest second.

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

Q5:

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.

Q6:

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 centimetre.

Q7:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where π‘š ∠ 𝐢 = 6 2 ∘ and 𝐴 𝐢 = 1 7 c m . Find the lengths of 𝐴 𝐡 and 𝐡 𝐢 giving the answer to two decimal places and the measure of ∠ 𝐴 giving the answer to the nearest degree.

  • A 𝐴 𝐡 = 1 5 . 0 1 c m , 𝐡 𝐢 = 7 . 9 8 c m , π‘š ∠ 𝐴 = 2 8 ∘
  • B 𝐴 𝐡 = 7 . 9 8 c m , 𝐡 𝐢 = 1 5 . 0 1 c m , π‘š ∠ 𝐴 = 2 8 ∘
  • C 𝐴 𝐡 = 7 . 9 8 c m , 𝐡 𝐢 = 1 5 . 0 1 c m , π‘š ∠ 𝐴 = 3 8 ∘
  • D 𝐴 𝐡 = 1 5 . 0 1 c m , 𝐡 𝐢 = 7 . 9 8 c m , π‘š ∠ 𝐴 = 3 8 ∘

Q8:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where 𝐴 𝐡 = 2 7 c m and π‘š ∠ 𝐴 = 6 3 ∘ . Find the lengths 𝐴 𝐢 and 𝐡 𝐢 giving the answer to two decimal places and the measure of angle 𝐢 giving the answer to the nearest degree.

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

Q9:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where π‘š ∠ 𝐢 = 1 . 1 8 8 r a d and 𝐴 𝐢 = 1 2 c m . Find π‘š ∠ 𝐴 in radians and lengths 𝐴 𝐡 and 𝐡 𝐢 giving all answers to three decimal places.

  • A π‘š ∠ 𝐴 = 0 . 3 8 3 r a d , 𝐴 𝐡 = 1 1 . 1 3 1 c m , 𝐡 𝐢 = 4 . 4 8 2 c m
  • B π‘š ∠ 𝐴 = 0 . 3 8 3 r a d , 𝐴 𝐡 = 1 1 . 1 3 1 c m , 𝐡 𝐢 = 2 9 . 8 0 2 c m
  • C π‘š ∠ 𝐴 = 1 . 1 8 8 r a d , 𝐴 𝐡 = 1 1 . 1 3 1 c m , 𝐡 𝐢 = 1 1 . 1 3 1 c m
  • D π‘š ∠ 𝐴 = 0 . 5 5 7 r a d , 𝐴 𝐡 = 4 . 4 8 2 c m , 𝐡 𝐢 = 1 1 . 1 3 1 c m

Q10:

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.

Q11:

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 centimetre.

Q12:

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

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

Q13:

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.

Q14:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where π‘š ∠ 𝐴 = 0 . 9 4 2 r a d and 𝐴 𝐡 = 1 9 c m . Find π‘š ∠ 𝐢 in radians and the lengths 𝐡 𝐢 and 𝐴 𝐢 giving all answers to three decimal places.

  • A π‘š ∠ 𝐢 = 0 . 6 2 9 r a d , 𝐡 𝐢 = 2 6 . 1 2 5 c m , 𝐴 𝐢 = 3 2 . 3 0 3 c m
  • B π‘š ∠ 𝐢 = 0 . 6 2 9 r a d , 𝐡 𝐢 = 1 1 . 1 7 5 c m , 𝐴 𝐢 = 3 2 . 3 0 3 c m
  • C π‘š ∠ 𝐢 = 2 . 2 0 0 r a d , 𝐡 𝐢 = 3 2 . 3 0 3 c m , 𝐴 𝐢 = 2 6 . 1 2 5 c m
  • D π‘š ∠ 𝐢 = 2 . 2 0 0 r a d , 𝐡 𝐢 = 2 6 . 1 2 5 c m , 𝐴 𝐢 = 2 3 . 4 9 3 c m

Q15:

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.

Q16:

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.

Q17:

𝐴 𝐡 𝐢 𝐷 is a trapezium where 𝐴 𝐷 βˆ₯ 𝐡 𝐢 , 𝐡 𝐢 = 2 0 c m , π‘š ∠ 𝐴 𝐡 𝐢 = 6 1 ∘ , 𝐴 𝐷 = 1 1 c m and π‘š ∠ 𝐡 𝐴 𝐢 = 5 9 ∘ . Find the area of the trapezium giving the answer to the nearest square centimetre.

Q18:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž = 1 9 cm, 𝑏 = 9 cm, and π‘š ∠ 𝐢 = 4 5 ∘ . Find the radius of the circumcircle giving the answer to two decimal places.

Q19:

𝐴 𝐡 𝐢 𝐷 is a trapezium, where 𝐴 𝐷 βˆ₯ 𝐡 𝐢 , 𝐴 𝐷 = 2 0 c m , π‘š ∠ 𝐡 = 5 5 ∘ , π‘š ∠ 𝐷 = 8 0 ∘ , and π‘š ∠ 𝐴 𝐢 𝐡 = 5 2 ∘ . Find the area of the trapezium giving the answer to the nearest square centimetre.

Q20:

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.

Q21:

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 4 5 1 6 β€² 3 0 β€² β€² ∘ , 67.142 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 1 1 1 1 0 β€² 6 β€² β€² ∘ , 88.123 cm2

Q22:

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 3 0 3 3 β€² 3 0 β€² β€² ∘ , 178.454 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 7 0 4 6 β€² 1 1 β€² β€² ∘ , 331.415 cm2

Q23:

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 3 1 0 β€² 2 5 β€² β€² ∘ , 51.963 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 2 8 3 6 β€² 4 8 β€² β€² ∘ , 178.154 cm2

Q24:

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 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 ∘ ∘ ∘
  • 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 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 ∘ ∘ ∘
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