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?

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

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the triangles DBC and CAD are congruent the location of A is justified

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AM is the altitude for the triangle ADC and hence the location of A is justified

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None of these

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Solution

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 $∠$ AMD = $∠$ AMC = $90^{\circ}$

In $△$ AMD and $△$ AMC,

MD = MC (perpendicular bisector bisects the line DC)

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

AM = AM (common side)

$△$ AMD $≅$ $△$ 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.

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Similar questions

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?

Pinku was a hard working student who used to mug up without understanding. He was asked to construct a triangle say ABC and was given details like the base length of the triangle BC, one of the base angles say $∠$ B and the sum of the other two sides (AB+AC). He went about constructing the triangle in the following way:

He drew the base BC with the given dimension, drew the $∠$ B along the ray BX with the angle known to him already. He then took B as centre and (AB + AC) as radius and cuts an arc on the ray BX intersecting the ray at D.

He then joins D to C. He then draws a perpendicular bisector of the line DC and the perpendicular bisector intersecting on the ray intersects the ray at point A. The teacher then asked him as to why he did what he did, she retracked the steps and asked him as to how the intersection of the ray and the perpendicular bisector gives A.

Which of the following is the reason for him drawing the perpendicular bisector and intersecting it with the ray?

Pinku was a hard working student who used to learn without understanding. He was asked to construct a triangle say ABC and was given the base length of the triangle BC, one of the base angles say $∠$ B and the sum of the other two sides (AB + AC). He went about constructing the triangle in the following way:

He then joins D to C. He then draws a perpendicular bisector of the line DC and the perpendicular bisector intersecting on the ray intersects the ray at point A. The teacher then asked him as to why he did what he did, she started from the back and asked him as to how the intersection of the ray and the perpendicular bisector gives A.

Which of the following is the reason for him drawing the perpendicular bisector and intersecting it with the ray?

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 $∠$ 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?

The steps to draw a triangle with the base BC, a base angle $∠$ B and the difference of other two sides is given below.
1. Draw the base BC and at point B make an angle say XBC equal to the given angle.
2. Cut the line segment BD equal to AB - AC on the reflection of ray BX (i.e. BX').
3. Join DC and draw the perpendicular bisector, say PQ of DC.
The next step will be:

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