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Worksheet: Angle Bisector Theorem

Q1:

In the given figure, 𝐴 𝐡 = 3 5 , 𝐴 𝐢 = 3 0 , and 𝐢 𝐷 = 1 2 . If 𝐡 𝐷 = π‘₯ + 1 0 , what is the value of π‘₯ ?

Q2:

If β–³ 𝐡 𝐴 𝐷 is a right-angled triangle at 𝐴 , 𝐴 𝐢 = 1 0 c m , 𝐢 𝐸 = 1 2 c m , and 𝐸 𝐴 = 1 5 c m , calculate the value of π‘₯ .

Q3:

Given that 𝐴 𝐡 = 6 0 , 𝐴 𝐢 = 4 0 , and 𝐡 𝐢 = 3 1 , what is 𝐢 𝐷 ?

Q4:

Given that 𝐴 𝐡 = 5 4 , 𝐴 𝐢 = 3 4 , and 𝐡 𝐢 = 3 0 , what is 𝐢 𝐷 ?

Q5:

In the triangle 𝐴 𝐡 𝐢 , 𝐴 𝐡 = 7 6 c m , 𝐴 𝐢 = 5 7 c m , and 𝐡 𝐷 = 5 2 c m . Given that 𝐴 𝐷 bisects ∠ 𝐴 and intersects 𝐡 𝐢 at 𝐷 , determine the length of 𝐴 𝐷 .

Q6:

In the given figure, if 𝐴 𝐡 ∢ 𝐴 𝐢 ∢ 𝐡 𝐢 = 6 ∢ 9 ∢ 1 1 , find 𝐡 𝐷 ∢ 𝐷 𝐢 .

  • A 9 ∢ 1 1
  • B 6 ∢ 1 1
  • C 3 ∢ 2
  • D 2 ∢ 3

Q7:

If 𝐴 𝐡 ∢ 𝐴 𝐢 = 3 ∢ 5 and 𝐡 𝐷 = 2 7 c m , determine the perimeter of β–³ 𝐴 𝐡 𝐢 .

Q8:

If 𝐴 𝐡 = 2 5 c m and 𝐴 𝐢 = 2 1 c m , find 𝐡 𝐸 𝐡 𝐢 . Leave your answer as a fraction in its simplest form.

  • A 2 5 2 1
  • B 4 6 2 5
  • C 2 1 2 5
  • D 2 5 4 6

Q9:

Suppose that in the figure, . What is ?

  • A
  • B
  • C
  • D

Q10:

Given that in the figure, 𝐴 𝐡 = 8 , 𝐡 𝐢 = 1 5 , and 𝐴 𝐢 = 2 0 , what is 𝐸 𝐡 ?

Q11:

Using the figure below, find the length of 𝐴 𝐷 to the nearest hundredth.

Q12:

𝐴 𝐡 is a chord in a circle. 𝐷 ∈ major arc 𝐴 𝐡 such that 𝐴 𝐷 𝐷 𝐡 = 1 2 . 𝐸 is the midpoint of the minor arc 𝐴 𝐡 . 𝐷 𝐸 is drawn to intersect 𝐴 𝐡 at 𝐢 . Determine the ratio between the areas of β–³ 𝐴 𝐷 𝐸 and β–³ 𝐡 𝐷 𝐸 .

  • A1
  • B2
  • C4
  • D 1 2

Q13:

𝐴 𝐡 𝐢 is a triangle in which 𝐴 𝐡 = 3 2 c m , 𝐡 𝐢 = 3 3 c m , and 𝐴 𝐢 = 1 6 c m . 𝐷 ∈ 𝐡 𝐢 , where 𝐡 𝐷 = 2 2 c m , 𝐴 𝐸 βŸ‚ 𝐴 𝐷 and intersects 𝐡 𝐢 at 𝐸 . If 𝐴 𝐷 bisects ∠ 𝐡 𝐴 𝐢 , find the length of 𝐢 𝐸 .

Q14:

In the figure,  𝐴 𝐷 bisects ∠ 𝐡 𝐴 𝐢 , 𝐡 𝐷 = 8 , 𝐷 𝐢 = 1 1 , and the perimeter of β–³ 𝐴 𝐡 𝐢 is 57. Determine the lengths of 𝐴 𝐡 and 𝐴 𝐢 .

  • A 𝐴 𝐡 = 1 6 , 𝐴 𝐢 = 1 9
  • B 𝐴 𝐡 = 1 9 , 𝐴 𝐢 = 2 2
  • C 𝐴 𝐡 = 2 2 , 𝐴 𝐢 = 1 6
  • D 𝐴 𝐡 = 1 6 , 𝐴 𝐢 = 2 2

Q15:

In triangle 𝐴 𝐡 𝐢 , 𝐷 lies on 𝐴 𝐢 such that  𝐡 𝐷 bisects ∠ 𝐴 𝐡 𝐢 . Given 𝐴 𝐡 = 1 0 , 𝐡 𝐢 = 2 0 , and 𝐴 𝐷 = 6 , determine 𝐴 𝐢 to the nearest hundredth.

Q16:

If 𝐴 𝐡 = 3 0 c m , 𝐡 𝐢 = 4 0 c m , and 𝐴 𝐢 = 4 5 c m , find the ratio between the areas of the β–³ 𝐴 𝐸 𝐷 and the β–³ 𝐴 𝐸 𝐢 .

  • A 3 ∢ 4
  • B 8 ∢ 9
  • C 8 ∢ 1 5
  • D 4 ∢ 5

Q17:

Given that 𝐴 𝐡 = π‘₯ + 5 c m , 𝐴 𝐢 = 2 9 c m , 𝐢 𝐷 = 3 8 c m , and 𝐡 𝐢 = 3 8 c m , find the numerical value of π‘₯ .

Q18:

Find the lengths of 𝐴 𝐢 and 𝐴 𝐷 in the figure.

  • A 𝐴 𝐢 = 5 5 c m , 𝐴 𝐷 = 5 8 c m
  • B 𝐴 𝐢 = 5 8 c m , 𝐴 𝐷 = 5 5 c m
  • C 𝐴 𝐢 = 5 5 c m , 𝐴 𝐷 = 5 0 c m
  • D 𝐴 𝐢 = 5 0 c m , 𝐴 𝐷 = 4 0 c m

Q19:

In the shown figure, determine 𝐴 𝐷 ∢ 𝐡 𝐷 .

  • A 9 ∢ 7
  • B 1 6 ∢ 9
  • C 7 ∢ 9
  • D 9 ∢ 1 6

Q20:

Use the figure to determine length 𝐴 𝐷 to two decimal places.

Q21:

𝐴 𝐡 𝐢 is a triangle, where 𝑋 is the midpoint of 𝐡 𝐢 , 𝐡 𝑋 = 2 3 c m and 𝐴 𝑋 = 2 3 c m . If the bisector of ∠ 𝐴 𝑋 𝐡 intersects 𝐴 𝐡 at 𝐷 , find the value of 𝐴 𝐷 𝐷 𝐡 .

  • A 2 7 4 6
  • B 2 3 2 7
  • C 4 6 2 7
  • D 2 7 2 3

Q22:

Given that angle 𝐴 is bisected by 𝐷 𝐴 , 𝐴 𝐡 = 3 8 , 𝐴 𝐢 = 1 8 , and 𝐡 𝐢 = 2 8 . Determine the lengths 𝐷 𝐡 and 𝐷 𝐢 .

  • A 𝐷 𝐡 = 9 , 𝐷 𝐢 = 1 9
  • B 𝐷 𝐡 = 1 4 , 𝐷 𝐢 = 1 4
  • C 𝐷 𝐡 = 1 2 , 𝐷 𝐢 = 1 6
  • D 𝐷 𝐡 = 1 9 , 𝐷 𝐢 = 9

Q23:

Given that 𝐴 𝐡 𝐢 is a triangle in which 𝐴 𝐢 = 1 0 c m , find the value of each of π‘₯ and 𝑦 .

  • A π‘₯ = √ 6 6 , 𝑦 = 1 2
  • B π‘₯ = 8 , 𝑦 = √ 6 6
  • C π‘₯ = 1 2 , 𝑦 = √ 6 6
  • D π‘₯ = √ 6 6 , 𝑦 = 8

Q24:

is a right triangle at , where bisects and intersects at . Given that , , determine the perimeter of .

  • A 27 cm
  • B 139.5 cm
  • C 49.5 cm
  • D 162 cm

Q25:

If 𝐴 𝐡 𝐢 𝐷 is a quadrilateral in which 𝐴 𝐡 = 1 0 c m , 𝐡 𝐢 = 5 c m , 𝐢 𝐷 = 6 c m , 𝐴 𝐷 = 1 1 c m , where  𝐴 𝐸 bisects ∠ 𝐴 and 𝐡 𝐷 intersects at 𝐸 , find the value of the ratio 𝐡 𝐸 𝐸 𝐷 .

  • A 6 5
  • B 1 1 1 0
  • C 5 6
  • D 1 0 1 1