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Lesson: The Tangent Ratio

In this lesson, we will learn how to find an angle in a right triangle using the tangent ratio.

Video

12:43

Sample Question Videos

07:58
08:36
01:36

Worksheet • 25 Questions • 3 Videos

Q1:

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

  • A 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 ∘
  • 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 0 1 4 β€² 1 8 β€² β€² ∘ , 𝛽 = 5 4 2 0 β€² 4 1 β€² β€² ∘

Q2:

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 3 3 2 9 β€² 4 3 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 4 1 2 6 β€² 5 β€² β€² ∘

Q3:

𝐴 𝐡 𝐢 𝐷 is a rhombus where 𝐴 𝐢 = 2 4 c m and 𝐡 𝐷 = 3 2 c m . Find π‘š ∠ 𝐡 𝐴 𝐢 giving the answer to the nearest second.

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

Q4:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where 𝐴 𝐡 = 1 9 c m and π‘š ∠ 𝐢 = 5 4 ∘ . Find the length of 𝐡 𝐢 giving the answer to two decimal places.

Q5:

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 0 2 9 β€² 4 0 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 6 4 β€² 4 8 β€² β€² ∘

Q6:

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 8 2 5 2 β€² 3 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 7 5 5 7 β€² 5 0 β€² β€² ∘

Q7:

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

  • A
  • B
  • C
  • D

Q8:

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

Q9:

Find π‘₯ to two decimal places.

Q10:

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

Q11:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 , where 𝐴 𝐡 = 6 4 c m . Point 𝐷 lies on 𝐡 𝐢 and point 𝐸 lies on 𝐴 𝐢 where 𝐸 𝐷 βˆ₯ 𝐴 𝐡 . Find the length of 𝐷 𝐡 given 𝐢 𝐷 = 1 2 c m and 𝐸 𝐷 = 3 5 c m . Give the answer to two decimal places.

Q12:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 where π‘š ∠ 𝐢 = 3 4 1 2 β€² ∘ and 𝐡 𝐢 = 2 5 c m . Find the length of 𝐴 𝐡 giving the answer to two decimal places.

Q13:

Find the exact value of t a n 𝐢 .

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

Q14:

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 1 4 2 β€² 1 0 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 4 7 1 7 β€² 2 6 β€² β€² ∘

Q15:

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.

Q16:

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

Q17:

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 centimetre, the length of 𝐡 𝐢 .

Q18:

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

Q19:

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

Q20:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡 . Point 𝐷 lies on 𝐡 𝐢 and point 𝐸 lies on 𝐴 𝐢 where 𝐸 𝐷 βˆ₯ 𝐴 𝐡 . Find the area of the trapezium 𝐴 𝐡 𝐷 𝐸 given 𝐴 𝐡 = 2 4 c m , 𝐢 𝐷 = 3 5 c m , and 𝐸 𝐷 = 1 2 c m . Give the answer to two decimal places.

Q21:

𝐴 𝐡 𝐢 is a right angled triangle at 𝐡 . Find the area of 𝐴 𝐡 𝐢 given that 𝐴 𝐡 = 2 4 c m , 𝐷 ∈ 𝐴 𝐢 , 𝐸 ∈ 𝐡 𝐢 , 𝐷 𝐸 βŸ‚ 𝐡 𝐢 , and 5 𝐷 𝐸 = 6 𝐸 𝐢 . Round your answer to two decimal places.

Q22:

Find the value of t a n t a n 𝐡 𝐢 given 𝐴 𝐡 𝐢 is an isosceles triangle where 𝐴 𝐡 = 𝐴 𝐢 = 4 5 c m and 𝐡 𝐢 = 7 2 c m .

  • A 9 1 6
  • B 1 5 1 6
  • C 6 4 9
  • D 1 6 9

Q23:

𝐴 𝐡 𝐢 is a triangle where 𝑏 = 1 3 √ 3 π‘Ž and π‘š ∠ 𝐢 = 1 5 0 ∘ . Find the value of t a n 𝐴 without using a calculator.

  • A 1 2 7 √ 3
  • B 2 7 √ 3
  • C 2 5 √ 3
  • D 1 2 5 √ 3

Q24:

In the given figure, the two triangles are similar.

Work out the value of t a n πœƒ for β–³ 𝐴 𝐡 𝐢 . Give your answer as a fraction in its simplest form.

  • A 1 1 1 2
  • B 1 1 √ 2 6 5 2 6 5
  • C 1 2 1 1
  • D √ 2 6 5 1 1
  • E 1 2 √ 2 6 5 2 6 5

Work out the value of t a n πœƒ for β–³ 𝐷 𝐸 𝐹 . Give your answer as a fraction in its simplest form.

  • A 1 1 1 2
  • B 1 2 √ 2 6 5 2 6 5
  • C √ 2 6 5 1 2
  • D 1 1 √ 2 6 5 2 6 5
  • E 1 2 1 1

What can be said about the value of t a n πœƒ for two similar triangles?

  • AThey are often equal.
  • BThey are always equal.
  • CThey will never be equal.

Q25:

𝐴 𝐡 is a diametre of a circle with radius 17 cm. Point 𝐢 is on the circumference of the circle where 𝐴 𝐢 βŠ₯ 𝐢 𝐡 and 𝐴 𝐢 = 1 6 c m . Find the exact values of t a n 𝐴 and t a n 𝐡 .

  • A t a n 𝐴 = 1 5 8 , t a n 𝐡 = 8 1 5
  • B t a n 𝐴 = 8 1 5 , t a n 𝐡 = 1 5 8
  • C t a n 𝐴 = 8 1 7 , t a n 𝐡 = 1 5 8
  • D t a n 𝐴 = 1 5 8 , t a n 𝐡 = 8 1 7
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