Worksheet: Perpendicular Bisector Theorem and Its Converse

In this worksheet, we will practice using the perpendicular bisector theorem and its converse to find a missing angle or side in an isosceles triangle.

Q1:

For which values of π‘₯ and 𝑦 is 𝐴𝐷 a perpendicular bisector of 𝐡𝐢?

  • A π‘₯ = 2 , 𝑦 = 7 5
  • B π‘₯ = 2 3 , 𝑦 = 1
  • C π‘₯ = 4 3 , 𝑦 = 3 5
  • D π‘₯ = 2 , 𝑦 = 1
  • E π‘₯ = 8 3 , 𝑦 = 7 5

Q2:

Determine whether 𝐴𝐸 is a perpendicular bisector of 𝐡𝐢.

  • AIt is a perpendicular bisector.
  • BIt is not a perpendicular bisector.

Q3:

In the diagram, 𝐴𝐡=6 and 𝐡𝐷=5.

Find 𝐴𝐢.

Find 𝐢𝐷.

Q4:

In the diagram, 𝐴𝐷 is the perpendicular bisector of 𝐡𝐢. Find the value of π‘₯.

Q5:

What makes an intersecting line a perpendicular bisector?

  • Awhen the intersecting line divides the other into two line segments of equal length
  • Bwhen the line intersects a line segment at an obtuse angle and divides it into two line segments of equal length
  • Cwhen the line intersects a line segment at right angles and divides it into two line segments of equal length
  • Dwhen the two lines meet at a right angle and the segment of each line is consequently of equal length
  • Ewhen the line intersects a line segment at an acute angle and divides it into two line segments of equal length

Q6:

When is a line said to be an angle bisector?

  • Awhen it divides an angle into two distinct angles
  • Bwhen it divides an angle into two angles of equal measure
  • Cwhen it cuts another line segment into two equal parts
  • Dwhen it connects two angles
  • Ewhen it cuts another line segment into two distinct parts

Q7:

In the following figure, find the length of π‘Šπ‘Œ.

Q8:

In the following figure, find the length of 𝐾𝐿.

Q9:

In the figure, what is the length of 𝐸𝐺?

Q10:

Find the length of 𝐴𝐢 and π‘šβˆ π·π΅πΆ.

  • A 116 cm, 6 0 ∘
  • B 58 cm, 4 5 ∘
  • C 116 cm, 4 5 ∘
  • D 58 cm, 3 0 ∘
  • E 116 cm, 3 0 ∘

Q11:

What is π‘šβˆ π‘‹πΆπ΅?

  • A 8 4 ∘
  • B 1 6 8 ∘
  • C 5 2 ∘
  • D 6 4 ∘

Q12:

The given figure shows an isosceles triangle, where 𝑀 is the midpoint of 𝐴𝐡.

Can we prove that triangle 𝐴𝐢𝑀 and triangle 𝐡𝐢𝑀 are congruent? If yes, state which congruence criteria could be used.

  • AYes, SAS
  • BNo
  • CYes, ASA
  • DYes, SSS

Hence, what can we conclude about the angles 𝐢𝐴𝐡 and 𝐴𝐡𝐢?

  • AThe angle 𝐢𝐴𝐡 is bigger than the angle 𝐴𝐡𝐢, because the two triangles are congruent.
  • BThe angle 𝐴𝐡𝐢 is bigger than the angle 𝐢𝐴𝐡, because the two triangles are congruent.
  • CThe angles have the same measure, because the triangles are congruent.
  • DWe cannot conclude anything, because we need more information.

Q13:

Find π‘šβˆ π·π΄π΅.

Q14:

In △𝐴𝐡𝐢, if 𝐴𝐡=𝐴𝐢 and π‘šβˆ π΄=52∘, find π‘šβˆ π΅.

Q15:

In the figure below, what is the area of β–³π‘‹π‘Œπ‘?

Q16:

Given that 𝐢𝐴=𝐢𝐷=𝐴𝐡, π‘šβˆ π·πΆπ΄=48∘, and π‘šβˆ π΄π΅πΆ=(3π‘₯+21)∘, find the value of π‘₯.

  • A18
  • B4
  • C13
  • D12
  • E26

Q17:

Find π‘₯.

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