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In this lesson, we will learn how to use the angle bisector theorem, its converse, and the incenter theorem to solve various problems.

Q1:

In the given figure, π΄ π΅ = 3 5 , π΄ πΆ = 3 0 , and πΆ π· = 1 2 . If π΅ π· = π₯ + 1 0 , what is the value of π₯ ?

Q2:

In the figure, ο« π΄ π· bisects β π΅ π΄ πΆ , π΅ π· = 8 , π· πΆ = 1 1 , and the perimeter of β³ π΄ π΅ πΆ is 57. Determine the lengths of π΄ π΅ and π΄ πΆ .

Q3:

Given that angle π΄ is bisected by π· π΄ , π΄ π΅ = 3 8 , π΄ πΆ = 1 8 , and π΅ πΆ = 2 8 . Determine the lengths π· π΅ and π· πΆ .

Q4:

If π΄ π΅ = 3 0 c m , π΅ πΆ = 4 0 c m , and π΄ πΆ = 4 5 c m , find the ratio between the areas of the β³ π΄ πΈ π· and the β³ π΄ πΈ πΆ .

Q5:

Find the lengths of π΄ πΆ and π΄ π· in the figure.

Q6:

If β³ π΅ π΄ π· is a right-angled triangle at π΄ , π΄ πΆ = 1 0 c m , πΆ πΈ = 1 2 c m , and πΈ π΄ = 1 5 c m , calculate the value of π₯ .

Q7:

If π΄ π΅ πΆ π· is a quadrilateral in which π΄ π΅ = 1 0 c m , π΅ πΆ = 5 c m , πΆ π· = 6 c m , π΄ π· = 1 1 c m , where ο« π΄ πΈ bisects β π΄ and π΅ π· intersects at πΈ , find the value of the ratio π΅ πΈ πΈ π· .

Q8:

Given that π΄ π΅ πΆ is a triangle in which π΄ πΆ = 1 0 c m , find the value of each of π₯ and π¦ .

Q9:

In the triangle π΄ π΅ πΆ , π΄ π΅ = 7 6 c m , π΄ πΆ = 5 7 c m , and π΅ π· = 5 2 c m . Given that π΄ π· bisects β π΄ and intersects π΅ πΆ at π· , determine the length of π΄ π· .

Q10:

Given that in the figure, π΄ π΅ = 8 , π΅ πΆ = 1 5 , and π΄ πΆ = 2 0 , what is πΈ π΅ ?

Q11:

If π΄ π΅ βΆ π΄ πΆ = 3 βΆ 5 and π΅ π· = 2 7 c m , determine the perimeter of β³ π΄ π΅ πΆ .

Q12:

Using the figure below, find the length of π΄ π· to the nearest hundredth.

Q13:

In triangle π΄ π΅ πΆ , π· lies on π΄ πΆ such that ο« π΅ π· bisects β π΄ π΅ πΆ . Given π΄ π΅ = 1 0 , π΅ πΆ = 2 0 , and π΄ π· = 6 , determine π΄ πΆ to the nearest hundredth.

Q14:

Use the figure to determine length π΄ π· to two decimal places.

Q15:

Given that π΄ π΅ = 6 0 , π΄ πΆ = 4 0 , and π΅ πΆ = 3 1 , what is πΆ π· ?

Q16:

π΄ π΅ πΆ is a right-angled triangle at π΅ , where ο« π΄ π· bisects β π΄ and intersects π΅ πΆ at π· . Given that π΅ π· = 1 8 c m , π΅ π΄ βΆ π΄ πΆ = 4 βΆ 5 , determine the perimeter of β³ π΄ π΅ πΆ .

Q17:

In the given figure, if π΄ π΅ βΆ π΄ πΆ βΆ π΅ πΆ = 6 βΆ 9 βΆ 1 1 , find π΅ π· βΆ π· πΆ .

Q18:

If π΄ π΅ = 2 5 c m and π΄ πΆ = 2 1 c m , find π΅ πΈ π΅ πΆ . Leave your answer as a fraction in its simplest form.

Q19:

Suppose that in the figure, π β π· π΄ πΆ = 3 4 β . What is π β πΈ π΄ πΉ ?

Q20:

π΄ π΅ is a chord in a circle. π· β major arc π΄ π΅ such that π΄ π· π· π΅ = 1 2 . πΈ is the midpoint of the minor arc π΄ π΅ . π· πΈ is drawn to intersect π΄ π΅ at πΆ . Determine the ratio between the areas of β³ π΄ π· πΈ and β³ π΅ π· πΈ .

Q21:

π΄ π΅ πΆ is a triangle in which π΄ π΅ = 3 2 c m , π΅ πΆ = 3 3 c m , and π΄ πΆ = 1 6 c m . π· β π΅ πΆ , where π΅ π· = 2 2 c m , π΄ πΈ β π΄ π· and intersects π΅ πΆ at πΈ . If π΄ π· bisects β π΅ π΄ πΆ , find the length of πΆ πΈ .

Q22:

Given that π΄ π΅ = π₯ + 5 c m , π΄ πΆ = 2 9 c m , πΆ π· = 3 8 c m , and π΅ πΆ = 3 8 c m , find the numerical value of π₯ .

Q23:

In the shown figure, determine π΄ π· βΆ π΅ π· .

Q24:

π΄ π΅ πΆ is a triangle, where π is the midpoint of π΅ πΆ , π΅ π = 2 3 c m and π΄ π = 2 3 c m . If the bisector of β π΄ π π΅ intersects π΄ π΅ at π· , find the value of π΄ π· π· π΅ .

Q25:

In the figure, π΄ π΅ βΆ π΄ πΆ = 4 βΆ 7 . What is π΅ π· βΆ π΅ πΆ ?

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