Point P is a point inside triangle ABC. Is triangle ABC equilateral?

Question

Triangle with interior point.png Point P is a point inside triangle ABC. Is triangle ABC equilateral? Statement #1 : Point P is equidistant from the three vertices A, B, and C. Statement #2 : Triangle ABC has two different lines of symmetry that pass through point P. Geometry is a beautiful topic! This question comes from a bank of 10 DS practice questions on Geometry. To see the others as well as the OE for this particular question, see: GMAT Data Sufficiency Geometry Practice Questions Mike :-)

chetan2u · Answer

Hi,

lets see the statements- 
 Statement #1 : Point P is equidistant from the three vertices A, B, and C.
This is same as THREE points on a circle and point P is the CENTER of the circle...
here we can have various combination of picking 3 points on circle and most will not make an equilateral triangle..
ans can be YES and NO both..
Insuff

Statement #2 : Triangle ABC has two different lines of symmetry that pass through point P.
for symmetry, the line has to pass through one vertice and the center of opposite side...
Only 1 such line would most likely make an isosceles triangle, BUT two such lines will happen only in equilateral triangle..
Suff

B

barryseal · Answer

Do we need to know what lines of symmetry means? Never heard that term before in any gmat geometry question

Debashis Roy · Answer

@chetan2uHi@ chetan2u can you please elaborate this statement:

Statement #2: Triangle ABC has two different lines of symmetry that pass through point P.
for symmetry, the line has to pass through one vertice and the center of opposite side...
Only 1 such line would most likely make an isosceles triangle, BUT two such lines will happen only in equilateral triangle..

Thanks in advance

chetan2u · Answer

Line of symmetry means the line that will result in a mirror image on either side...
so for an isosceles triangle, there will be only one line of symmetry as shown in the figure. One line of symmetry would mean any two angles equal

However only for equilateral it will be more than 1, as it would mean angle A= B, and B=C, which would mean A=B=C so all angles equal.

Debashis Roy · Answer

chetan2u  Yeah I got it. Thanks.

Posted from my mobile device

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Point P is a point inside triangle ABC. Is triangle ABC equilateral?

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