Worksheet: Angle Bisector Theorem

In this worksheet, we will practice using the angle bisector theorem, its converse, and the incenter theorem to solve various problems.

Q1:

In the given figure, 𝐴 𝐵 = 3 5 , 𝐴 𝐶 = 3 0 , and 𝐶 𝐷 = 1 2 . If 𝐵 𝐷 = 𝑥 + 1 0 , what is the value of 𝑥 ?

Q2:

In the figure, 𝐴 𝐷 bisects 𝐵 𝐴 𝐶 , 𝐵 𝐷 = 8 , 𝐷 𝐶 = 1 1 , and the perimeter of 𝐴 𝐵 𝐶 is 57. Determine the lengths of 𝐴 𝐵 and 𝐴 𝐶 .

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

Q3:

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

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

Q4:

If 𝐴 𝐵 = 3 0 c m , 𝐵 𝐶 = 4 0 c m , and 𝐴 𝐶 = 4 5 c m , find the ratio between the areas of the 𝐴 𝐸 𝐷 and the 𝐴 𝐸 𝐶 .

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

Q5:

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

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

Q6:

If 𝐵 𝐴 𝐷 is a right-angled triangle at 𝐴 , 𝐴 𝐶 = 1 0 c m , 𝐶 𝐸 = 1 2 c m , and 𝐸 𝐴 = 1 5 c m , calculate the value of 𝑥 .

Q7:

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

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

Q8:

Given that 𝐴 𝐵 𝐶 is a triangle in which 𝐴 𝐶 = 1 0 c m , find the value of each of 𝑥 and 𝑦 .

  • A 𝑥 = 6 6 , 𝑦 = 1 2
  • B 𝑥 = 8 , 𝑦 = 6 6
  • C 𝑥 = 1 2 , 𝑦 = 6 6
  • D 𝑥 = 6 6 , 𝑦 = 8

Q9:

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

Q10:

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

Q11:

If 𝐴 𝐵 𝐴 𝐶 = 3 5 and 𝐵 𝐷 = 2 7 c m , determine the perimeter of 𝐴 𝐵 𝐶 .

Q12:

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

Q13:

In triangle 𝐴 𝐵 𝐶 , 𝐷 lies on 𝐴 𝐶 such that 𝐵 𝐷 bisects 𝐴 𝐵 𝐶 . Given 𝐴 𝐵 = 1 0 , 𝐵 𝐶 = 2 0 , and 𝐴 𝐷 = 6 , determine 𝐴 𝐶 to the nearest hundredth.

Q14:

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

Q15:

Given that 𝐴 𝐵 = 6 0 , 𝐴 𝐶 = 4 0 , and 𝐵 𝐶 = 3 1 , what is 𝐶 𝐷 ?

Q16:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 , where 𝐴 𝐷 bisects 𝐴 and intersects 𝐵 𝐶 at 𝐷 . Given that 𝐵 𝐷 = 1 8 c m , 𝐵 𝐴 𝐴 𝐶 = 4 5 , determine the perimeter of 𝐴 𝐵 𝐶 .

Q17:

In the given figure, if 𝐴 𝐵 𝐴 𝐶 𝐵 𝐶 = 6 9 1 1 , find 𝐵 𝐷 𝐷 𝐶 .

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

Q18:

If 𝐴 𝐵 = 2 5 c m and 𝐴 𝐶 = 2 1 c m , find 𝐵 𝐸 𝐵 𝐶 . 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

Q19:

Suppose that in the figure, 𝑚 𝐷 𝐴 𝐶 = 3 4 . What is 𝑚 𝐸 𝐴 𝐹 ?

Q20:

𝐴 𝐵 is a chord in a circle. 𝐷 major arc 𝐴 𝐵 such that 𝐴 𝐷 𝐷 𝐵 = 1 2 . 𝐸 is the midpoint of the minor arc 𝐴 𝐵 . 𝐷 𝐸 is drawn to intersect 𝐴 𝐵 at 𝐶 . Determine the ratio between the areas of 𝐴 𝐷 𝐸 and 𝐵 𝐷 𝐸 .

  • A1
  • B2
  • C4
  • D 1 2

Q21:

𝐴 𝐵 𝐶 is a triangle in which 𝐴 𝐵 = 3 2 c m , 𝐵 𝐶 = 3 3 c m , and 𝐴 𝐶 = 1 6 c m . 𝐷 𝐵 𝐶 , where 𝐵 𝐷 = 2 2 c m , 𝐴 𝐸 𝐴 𝐷 and intersects 𝐵 𝐶 at 𝐸 . If 𝐴 𝐷 bisects 𝐵 𝐴 𝐶 , find the length of 𝐶 𝐸 .

Q22:

Given that 𝐴 𝐵 = 𝑥 + 5 c m , 𝐴 𝐶 = 2 9 c m , 𝐶 𝐷 = 3 8 c m , and 𝐵 𝐶 = 3 8 c m , find the numerical value of 𝑥 .

Q23:

In the shown figure, determine 𝐴 𝐷 𝐵 𝐷 .

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

Q24:

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

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

Q25:

In the figure, 𝐴 𝐵 𝐴 𝐶 = 4 7 . What is 𝐵 𝐷 𝐵 𝐶 ?

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

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