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Worksheet: The Tangent Ratio

In this worksheet, we will practice finding an angle in a right triangle using the tangent ratio.

Q1:

Find the value of 𝐾 , given 𝐴 𝐡 𝐢 is an equilateral triangle, where point 𝐷 lies on 𝐴 𝐡 , 𝐴 𝐷 = 5 c m , 𝐷 𝐡 = 1 2 c m , and 𝐾 𝑋 = √ 3 t a n .

  • A 1 7 1 2
  • B 6 1 1
  • C 1 2 1 7
  • D 1 1 6

Q2:

Find π‘₯ to two decimal places.

Q3:

In the given figure, π‘š ∠ 𝐡 𝐴 𝐢 = 9 0 ∘ and 𝐴 𝐷 βŠ₯ 𝐡 𝐢 . What is 𝐴 𝐢 πœƒ t a n ?

  • A 𝐴 𝐷
  • B 𝐢 𝐷
  • C 𝐡 𝐢
  • D 𝐴 𝐡
  • E 𝐷 𝐡

Q4:

is a right triangle at , where . Point lies on and point lies on where . Find the length of given and . Give the answer to two decimal places.

  • A 33.94 cm
  • B 84.58 cm
  • C 174.67 cm
  • D 9.94 cm
  • E 29.24 cm

Q5:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 , where 𝐴 𝐡 = 4 5 c m and 𝐡 𝐢 = 6 8 c m . Find the measure of ∠ 𝐢 giving the answer to the nearest second.

  • A 4 1 2 6 β€² 5 β€² β€² ∘
  • B 4 8 3 3 β€² 5 5 β€² β€² ∘
  • C 5 6 3 0 β€² 1 7 β€² β€² ∘
  • D 3 3 2 9 β€² 4 3 β€² β€² ∘

Q6:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 , where 𝐴 𝐡 = 3 4 c m and 𝐡 𝐢 = 5 7 c m . Find the measure of ∠ 𝐴 giving the answer to the nearest second.

  • A 5 3 2 2 β€² 5 2 β€² β€² ∘
  • B 3 0 4 8 β€² 5 7 β€² β€² ∘
  • C 3 6 3 7 β€² 8 β€² β€² ∘
  • D 5 9 1 1 β€² 3 β€² β€² ∘

Q7:

is a right triangle at where and . Find the length of giving the answer to two decimal places.

  • A 8.19 cm
  • B 5.74 cm
  • C 19.28 cm
  • D 14.28 cm

Q8:

Find the exact value of t a n 𝐢 .

  • A 4 5
  • B 9 4
  • C 5 4
  • D 4 9

Q9:

is a right triangle at where and . Find the length of giving the answer to two decimal places.

  • A 23.49 cm
  • B 32.32 cm
  • C 11.17 cm
  • D 13.80 cm
  • E 26.15 cm

Q10:

is a right triangle at where and . Find the length of giving the answer to two decimal places.

  • A 14.59 cm
  • B 28.63 cm
  • C 5.90 cm
  • D 6.62 cm
  • E 25.51 cm

Q11:

The dimensions of a rectangular garden is 13 metres by 12 metres. Triangle 𝐴 𝐸 𝐢 represents the lawn where 𝐸 𝐢 = 1 0 m . Find the measure of ∠ 𝐸 𝐴 𝐡 giving the answer to the nearest second.

  • A 4 7 1 7 β€² 2 6 β€² β€² ∘
  • B 7 5 5 7 β€² 5 0 β€² β€² ∘
  • C 3 9 4 8 β€² 2 0 β€² β€² ∘
  • D 1 4 2 β€² 1 0 β€² β€² ∘

Q12:

𝑋 π‘Œ 𝑍 is a triangle where 𝐿 lies on 𝑋 𝑍 such that π‘Œ 𝐿 βŸ‚ 𝑋 𝑍 , t a n t a n 𝑍 = 𝑋 π‘Œ 𝐿 = 2 0 2 1 and 𝑍 𝐿 = 2 6 c m . Find the area of 𝑋 π‘Œ 𝑍 giving the answer to two decimal places.

Q13:

is a rhombus where and . Find giving the answer to the nearest second.

  • A
  • B
  • C
  • D

Q14:

Find the length of 𝐷 𝐢 given 𝐴 𝐡 𝐢 𝐷 is a quadrilateral where 𝐴 𝐷 = 6 c m and 𝐴 𝐡 = 8 c m .

  • A 4 4 5 cm
  • B 7 1 2 cm
  • C 10 cm
  • D 1 3 1 3 cm

Q15:

𝐴 𝐡 𝐢 is a triangle where 𝐴 𝐷 βŸ‚ 𝐡 𝐢 , 𝐴 𝐷 = 1 6 c m , π‘š ∠ 𝐡 = 4 4 ∘ , and π‘š ∠ 𝐢 = 4 8 ∘ . Find, to the nearest centimeter, the length of 𝐡 𝐢 .

  • A 45 cm
  • B 33 cm
  • C 46 cm
  • D 31 cm
  • E 23 cm

Q16:

In the given figure, π‘š ∠ 𝐡 𝐴 𝐢 = 9 0 ∘ where 𝐴 𝐷 βŠ₯ 𝐡 𝐢 . What is 𝐴 𝐷 πœƒ t a n ?

  • A 𝐴 𝐡
  • B 𝐡 𝐢
  • C 𝐴 𝐷
  • D 𝐡 𝐷
  • E 𝐴 𝐢

Q17:

Find the value of t a n t a n π‘₯ + 𝑦 given 𝐴 𝐡 𝐢 𝐷 is a square where points 𝐸 and 𝑂 lie on 𝐴 𝐢 , 𝐴 𝐸 = 8 c m , 𝐸 𝑂 = 2 5 c m , and 𝑂 𝐢 = 9 c m .

  • A 1 9 9 9 1
  • B 2 5 2 1
  • C 1 9 9 8 4
  • D 1 7 5 5 2

Q18:

is a right triangle at . Point lies on and point lies on where . Find the area of the trapezoid given , , and . Give the answer to two decimal places.

  • A
    1 890.00
    cm2
  • B 74.06 cm2
  • C 481.89 cm2
  • D 630.00 cm2

Q19:

Find the values of 𝛼 and 𝛽 giving your answer to the nearest second.

  • A 𝛼 = 3 0 1 4 β€² 1 8 β€² β€² ∘ , 𝛽 = 5 4 2 0 β€² 4 1 β€² β€² ∘
  • B 𝛼 = 5 4 2 0 β€² 4 1 β€² β€² ∘ , 𝛽 = 3 5 3 9 β€² 1 9 β€² β€² ∘
  • C 𝛼 = 5 4 2 0 β€² 4 1 β€² β€² ∘ , 𝛽 = 3 0 1 4 β€² 1 8 β€² β€² ∘
  • D 𝛼 = 3 5 3 9 β€² 1 9 β€² β€² ∘ , 𝛽 = 5 4 2 0 β€² 4 1 β€² β€² ∘

Q20:

Find the values of 𝛼 and 𝛽 giving your answer to the nearest second.

  • A 𝛼 = 4 0 3 6 β€² 4 6 β€² β€² ∘ , 𝛽 = 3 0 5 7 β€² 5 0 β€² β€² ∘
  • B 𝛼 = 3 0 5 7 β€² 5 0 β€² β€² ∘ , 𝛽 = 5 9 2 β€² 1 0 β€² β€² ∘
  • C 𝛼 = 3 0 5 7 β€² 5 0 β€² β€² ∘ , 𝛽 = 4 0 3 6 β€² 4 6 β€² β€² ∘
  • D 𝛼 = 5 9 2 β€² 1 0 β€² β€² ∘ , 𝛽 = 3 0 5 7 β€² 5 0 β€² β€² ∘

Q21:

𝐴 𝐡 𝐢 is a triangle where 𝐴 𝐷 βŸ‚ 𝐡 𝐢 , π‘š ∠ 𝐡 = 5 1 1 6 β€² 1 8 β€² β€² ∘ , 𝐡 𝐢 = 2 0 c m , and 𝐴 𝐷 = 8 c m . Find π‘š ∠ 𝐢 giving the answer to the nearest second.

  • A 3 6 4 β€² 4 8 β€² β€² ∘
  • B 5 3 5 5 β€² 1 2 β€² β€² ∘
  • C 1 6 5 0 β€² 5 6 β€² β€² ∘
  • D 3 0 2 9 β€² 4 0 β€² β€² ∘

Q22:

The following figure represents a line segment joining two points 𝐴 ( 2 , 0 ) and 𝐡 ( 3 , 8 ) . Find the measure of the angle πœƒ included between 𝐴 𝐡 and the π‘₯ -axis giving the answer to the nearest second.

  • A 7 5 5 7 β€² 5 0 β€² β€² ∘
  • B 2 6 3 3 β€² 5 4 β€² β€² ∘
  • C 6 9 2 6 β€² 3 8 β€² β€² ∘
  • D 8 2 5 2 β€² 3 0 β€² β€² ∘

Q23:

The following figure represents a line segment joining two points 𝐴 ( 4 , 0 ) and 𝐡 ( 9 , 7 ) . Find the measure of the angle πœƒ included between 𝐴 𝐡 and the π‘₯ -axis giving the answer to the nearest second.

  • A 6 0 1 5 β€² 1 8 β€² β€² ∘
  • B 7 1 3 3 β€² 5 4 β€² β€² ∘
  • C 3 7 5 2 β€² 3 0 β€² β€² ∘
  • D 5 4 2 7 β€² 4 4 β€² β€² ∘

Q24:

A flagpole 5.9 meters tall casts a 2.8-meter shadow. Find the angle of inclination of the sun giving the answer to the nearest minute.

  • A 2 8 2 0 ∘ β€²
  • B 6 4 3 7 ∘ β€²
  • C 6 1 4 0 ∘ β€²
  • D 2 5 2 3 ∘ β€²

Q25:

A flagpole 5.9 meters tall casts a 2.8-meter shadow. Find the angle of inclination of the sun giving the answer to the nearest minute.

  • A 5 6 9 ∘ β€²
  • B 5 0 1 7 ∘ β€²
  • C 3 3 5 1 ∘ β€²
  • D 3 9 4 3 ∘ β€²

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