In which of the following cases a unique triangle can be constructed?

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AB=4cm, BC=8cm, CA=2cm

An isosceles triangle in which the length of the equal sides is 6.2cm.

answer is C.

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Detailed Solution

Concept: If two triangles with the same parameters are congruent to one another, then the triangle with the specified parameters is unique. Determine which of the scenarios the triangle drawn will be unique using the fundamental congruence criteria like SSS, SAS, ASA, AAS, etc.
We determine which possibilities are correct option by option.
Two triangles will be drawn by us (One with the names of vertices as given and the other with vertex names as given added with a subscript). We determine which situations the triangles are congruent in and which situations the triangles might not be. So, we'll work out which scenario the triangle, if drawn, will be unique in. In order to determine if a triangle can be drawn or not, we will use the fact that the sum of two sides of a triangle is greater than the sum of the third side. The sum of the triangle's angles should be 180°.
Consider triangles

and

By SSS congruence

So, if the triangle does exist, it will be different. However, AB+AC = 4+2 = 6 <BC. As a result, using the provided dimensions, a triangle cannot be built.
Considering triangles

and

By ASA congruence rule

Therefore, the triangle will be unique if it does exist.
Though, we have

As a result, using the specified parameters, a triangle cannot be constructed.
Now consider

and

Let,

Thus by ASA congruence

So, if the triangle does exist, it will be unique.

Therefore, given the conditions, a triangle can be built. Two isosceles triangles with sides of 6.2 cm, 6.2 cm, 6.2 cm and 6.2 cm, 6.2 cm, 4 cm are being considered. It is obvious that both triangles satisfy option d. The triangles are not congruent, however, because the third side length varies between the two triangles. Additionally, because the sum of any two sides is greater than the sum of the third side, triangles with these side lengths can be created.
Hence, the correct option is 3.

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