Question

How will you prove that the construction for a triangle with the given conditions is right?

Given conditions: Base length BC is given, base angle B is given, and difference of the other two sides is given (AB-AC) where AB is greater than AC. For going about the construction, I drew the base length BC, drew the ray BX with angle XBC known to me. Taking B as centre and radius equal to (AB-AC) I cut an arc on the ray BX intersecting it at point D. I then joined D to C. Then drew the perpendicular bisector of the line segment DC and named the point of intersection of this perpendicular bisector and the ray BX as A. Joined A to C and the triangle ABC was ready

Which of the following statements gives the best explanation to this construction?

• A. Since the triangles AMD and AMC are congruent, AD = AC and hence the location of A has been plotted correctly
• B. the triangles DBC and CAD are congruent the location of A is justified
• C. AM is the altitude for the triangle ADC and hence the location of A is justified
• D. None of these

Answer 1

The correct option is A

Since the triangles AMD and AMC are congruent, AD = AC and hence the location of A has been plotted correctly

Since there was a perpendicular bisector drawn on the line DC, MD = MC and $\angle$AMD = $\angle$AMC = ${90}^{\circ }$

In $△$AMD and $△$AMC,

MD = MC (perpendicular bisector bisects the line DC)

$\angle$AMD = $\angle$AMC = ${90}^{\circ }$ (Perpendicular bisector)

AM = AM (common side)

$△$AMD $\cong$ $△$AMC (By SAS Congruency)

AD = AC (By CPCT)

Which means that since AB = BD + AD, and BD = AB-AC

This will only hold good if AD = AC which we just proved above which justifies the construction.