Question

Given are the steps of construction for constructing a triangle ABC whose base length is given say BC, one of the base angles is given, say $\angle$ B and the difference between the other two sides is also given (AB – AC) considering AB $>$ AC.

Pinku was asked to give the steps of construction for the same. He had mugged up all the steps of construction but forgot one point in between. Following are those steps of construction, let’s see if you can help Pinku with the missing step.

Steps of construction:

Draw the base BC and at point B make an angle say XBC equal to the given angle. Cut the line segment BD equal to (AB – AC) from ray BX. Let it intersect BX at a point A. Join AC

Which of the following do you think is the missing step of this construction?

• A. Join D to C and draw a perpendicular bisector of the line BC
• B. Join D to C and draw a perpendicular bisector of the line DC
• C. Join D to C and draw an angle bisector of the angle B
• D. Join D to C and draw a perpendicular bisector of the line AC

Answer 1

The correct option is B

Join D to C and draw a perpendicular bisector of the line DC

Pinku forgot the most important step of the construction where he has to Join the points D to C and then draw a perpendicular bisector of the line DC and make it intersect with the ray BX. The point of intersection of the perpendicular bisector and the ray gives us the point A, hence we are ready with the triangle ABC.

Why we do this is because if we take into consideration the $△$ AMD and $△$ AMC,

DM = MC (perpendicular bisector)

$\angle$AMD = $\angle$AMC = 90° (perpendicular bisector)

AM = AM (common side)

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

AD = AC (By CPCT)

This means that its like flipping the point A about the point D such that it merges with the point C.