A perpendicular bisector is a line that crosses (bisects) another line segment at 90 degrees, dividing the line segment it crosses into two equal length lines
The additional information Kiran need to know to show that AB is a perpendicular bisector is the fourth option;
DB and BC are congruent
The reason for the above selection is as follows:
The information about the isosceles triangle DAC Kiran knows are;
Triangle DAC = An isosceles triangle
Segment AD is congruent to segment AC
Segment AB is a is a line from the vertex point A to the side CD of triangle DAC
The required parameter:
The information Kiran has know so as to show that AB is a perpendicular bisector of CD
Method:
Determine an attribute present in triangle DAC with line AB drawn that is due to line AB being a perpendicular bisector
Solution:
For AB to be a perpendicular bisector, then DB and BC are congruent
Given that DB and BC are congruent, we get;
AB is congruent to AB by reflexive property
Therefore, given that AD is congruent to AC, we have;
Sides AD, AB, and DB in triangle ΔADB are congruent to the corresponding sides AC, BC, and AB in triangle ΔACB, which gives
ΔADB is congruent to ΔACB by Side-Side-Side, SSS, congruency rule
Therefore, ∠ABD ≅ ∠ABC, by Congruent Parts of Congruent Triangles are Congruent, CPCTC
∠ABD = ∠ABC by definition of congruency
∠ABD and ∠ABC are linear pair angles, therefore;
∠ABD + ∠ABC = 180°
∠ABD + ∠ABD = 2 × ∠ABD = 180° by substitution property
∠ABD = 180°/2 = 90°
∠ABD = 90° = ∠ABC, therefore, AB is perpendicular to CD and AB is a perpendicular bisector of CD
Therefore, the correct option Kiran need to know in order to show that AB is a perpendicular bisector of segment CD is the option; DB and BC are congruent
Learn more about perpendicular bisectors here:
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