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MULTIPLE TRIANGLES POSSIBLE

Constructions of Class 9

It is possible to draw more than one triangle that has three sides with the given lengths. For example in the figure below, given the base AB, you can draw four triangles that meet the requirements. All four are correct in that they satisfy the requirements, and are congruent to each other.

See figure on the right. If two sides add to less than the third, no triangle is possible.

Proof

The image below is the final drawing above with the red items added.

 

Argument

Reason

1

Line segment LM is congruent to AB.

Drawn with the same compass width. See Copying a line segment

2

The third vertex N of the triangle must lie somewhere on arc P.

All points on arc P are distance AC from L since the arc was drawn with the compass width set to AC.

3

The third vertex N of the triangle must lie somewhere on arc Q.

All points on arc Q are distance BC from M since the arc was drawn with the compass width set to BC.

4

The third vertex N must lie where the two arcs intersect

Only point that satisfies 2 and 3.

5

Triangle LMN satisfies the three side lengths given.
LM is congruent to AB,
LN is congruent to AC,
MN is congruent to BC,
 

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