Worksheet: Angle Bisector Theorem and Its Converse

In this worksheet, we will practice using the theorem of angle bisector and its converse to find a missing side length in a triangle.

Q1:

In the given figure, 𝐴𝐡=35, 𝐴𝐢=30, and 𝐢𝐷=12. If 𝐡𝐷=π‘₯+10, what is the value of π‘₯?

Q2:

In the figure, 𝐴𝐷 bisects ∠𝐡𝐴𝐢, 𝐡𝐷=8, 𝐷𝐢=11, and the perimeter of △𝐴𝐡𝐢 is 57. Determine the lengths of 𝐴𝐡 and 𝐴𝐢.

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

Q3:

Given that angle 𝐴 is bisected by 𝐷𝐴, 𝐴𝐡=38, 𝐴𝐢=18, and 𝐡𝐢=28. Determine the lengths 𝐷𝐡 and 𝐷𝐢.

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

Q4:

If 𝐴𝐡=30cm, 𝐡𝐢=40cm, and 𝐴𝐢=45cm, find the ratio between the areas of the △𝐴𝐸𝐷 and the △𝐴𝐸𝐢.

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

Q5:

Find the lengths of 𝐴𝐢 and 𝐴𝐷 in the figure.

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

Q6:

If △𝐡𝐴𝐷 is a right-angled triangle at 𝐴, 𝐴𝐢=10cm, 𝐢𝐸=12cm, and 𝐸𝐴=15cm, calculate the value of π‘₯.

Q7:

If 𝐴𝐡𝐢𝐷 is a quadrilateral in which 𝐴𝐡=10cm, 𝐡𝐢=5cm, 𝐢𝐷=6cm, 𝐴𝐷=11cm, where 𝐴𝐸 bisects ∠𝐴 and 𝐡𝐷 intersects at 𝐸, find the value of the ratio 𝐡𝐸𝐸𝐷.

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

Q8:

Given that 𝐴𝐡𝐢 is a triangle in which 𝐴𝐢=10cm, find the value of each of π‘₯ and 𝑦.

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

Q9:

In the triangle 𝐴𝐡𝐢, 𝐴𝐡=76cm, 𝐴𝐢=57cm, and 𝐡𝐷=52cm. Given that 𝐴𝐷 bisects ∠𝐴 and intersects 𝐡𝐢 at 𝐷, determine the length of 𝐴𝐷.

Q10:

Given that in the figure, 𝐴𝐡=8, 𝐡𝐢=15, and 𝐴𝐢=20, what is 𝐸𝐡?

Q11:

If 𝐴𝐡∢𝐴𝐢=3∢5 and 𝐡𝐷=27cm, determine the perimeter of △𝐴𝐡𝐢.

Q12:

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

Q13:

In triangle 𝐴𝐡𝐢, 𝐷 lies on 𝐴𝐢 such that 𝐡𝐷 bisects ∠𝐴𝐡𝐢. Given 𝐴𝐡=10, 𝐡𝐢=20, and 𝐴𝐷=6, determine 𝐴𝐢 to the nearest hundredth.

Q14:

Use the figure to determine length 𝐴𝐷 to two decimal places.

Q15:

Given that 𝐴𝐡=60, 𝐴𝐢=40, and 𝐡𝐢=31, what is 𝐢𝐷?

Q16:

𝐴 𝐡 𝐢 is a right-angled triangle at 𝐡, where 𝐴𝐷 bisects ∠𝐴 and intersects 𝐡𝐢 at 𝐷. Given that 𝐡𝐷=18cm, 𝐡𝐴∢𝐴𝐢=4∢5, determine the perimeter of △𝐴𝐡𝐢.

Q17:

In the given figure, if 𝐴𝐡∢𝐴𝐢∢𝐡𝐢=6∢9∢11, find 𝐡𝐷∢𝐷𝐢.

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

Q18:

If 𝐴𝐡=25cm and 𝐴𝐢=21cm, find 𝐡𝐸𝐡𝐢. Leave your answer as a fraction in its simplest form.

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

Q19:

Suppose that in the figure, π‘šβˆ π·π΄πΆ=34∘. What is π‘šβˆ πΈπ΄πΉ?

Q20:

𝐴 𝐡 is a chord in a circle. 𝐷∈ major arc 𝐴𝐡 such that 𝐴𝐷𝐷𝐡=12. 𝐸 is the midpoint of the minor arc 𝐴𝐡. 𝐷𝐸 is drawn to intersect 𝐴𝐡 at 𝐢. Determine the ratio between the areas of △𝐴𝐷𝐸 and △𝐡𝐷𝐸.

  • A4
  • B 1 2
  • C2
  • D1

Q21:

𝐴 𝐡 𝐢 is a triangle in which 𝐴𝐡=32cm, 𝐡𝐢=33cm, and 𝐴𝐢=16cm. 𝐷∈𝐡𝐢, where 𝐡𝐷=22cm, π΄πΈβŸ‚π΄π· and intersects 𝐡𝐢 at 𝐸. If 𝐴𝐷 bisects ∠𝐡𝐴𝐢, find the length of 𝐢𝐸.

Q22:

Given that 𝐴𝐡=π‘₯+5cm, 𝐴𝐢=29cm, 𝐢𝐷=38cm, and 𝐡𝐢=38cm, find the numerical value of π‘₯.

Q23:

In the shown figure, determine 𝐴𝐷∢𝐡𝐷.

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

Q24:

𝐴 𝐡 𝐢 is a triangle, where 𝑋 is the midpoint of 𝐡𝐢, 𝐡𝑋=23cm and 𝐴𝑋=23cm. If the bisector of βˆ π΄π‘‹π΅ intersects 𝐴𝐡 at 𝐷, find the value of 𝐴𝐷𝐷𝐡.

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

Q25:

In the figure, 𝐴𝐡∢𝐴𝐢=4∢7. What is 𝐡𝐷∢𝐡𝐢?

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.