State true or false.If an isosceles triangle has more than one line of symmetry then it is an equilateral triangle.

Question

Vedantu Content Team · Accepted Answer

Hint:Here we use the concept of symmetry which states that line of symmetry divides a figure into identical half so when we map the exact half we get the complete figure. Using the knowledge that the isosceles triangle has one line of symmetry we try to check for more than one line of symmetry.* An isosceles triangle is a triangle which has two sides of equal length and angles opposite to equal sides are also equal.Complete step-by-step answer:First we draw a diagram for one line of symmetry on an isosceles triangle.\n    \n    \n    \n    \n  Here $$\vartriangle ABC$$ is an isosceles triangle with sides $$AB = BC$$ and angles $$\angle B = \angle C$$.We have one line of symmetry $$n$$which divided the isosceles triangle into two exactly identical parts We can say that one triangle overlaps the other triangle if folded along the line of symmetry.Let us assume there is a second line of symmetry, be it from vertex $$C$$ to the side $$AB$$.We draw the diagram for such possibility \n    \n    \n    \n    \n  If $$m$$ is also a line of symmetry then folding the half side of the triangle should overlap the other half of the triangle if folded along the line $$m$$.So we can say that two triangles on either side of the line of symmetry are identical to each other Which gives that sides $$BC = AC$$ but we know $$\vartriangle ABC$$is an isosceles triangle so $$AB = AC$$Therefore two sides are equal to the same side means that all the three sides are equal to each other. $$AB = BC = AC$$Which is only possible if $$\vartriangle ABC$$ is an equilateral triangle which has all the three sides equal to each other.So, if an isosceles triangle has more than one line of symmetry then it is an equilateral triangle.Thus the statement is TRUE.Note:Students should always keep in mind that an equilateral triangle has three lines of symmetry each from one vertex to the opposite side of the triangle and the fact that isosceles triangle has only one line of symmetry.