Question 1210463
<pre>
I suspect that this problem was a mistranslation into English, as it could have
been mistranslated from a language where "centroid" could have translated as
"midpoint". Unlike tutor Ikleyn, I never write anything assuming the student did
anything wrong besides mistyping or mistranslating into English.  I think
perhaps the problem should have been stated this way:
 
A straight line segment L divides triangle D into two congruent triangles.
Select all the statements that must be true.

(a) If D is isosceles, then D is equilateral.
(b) If D is right, then D is equilateral.
(c) L is parallel to a side of D.
(d) L is longer than the distance from either endpoint of L to the centroid
    of D.

A straight line segment L divides triangle D into two congruent triangles. 

This means one end of L must be at a vertex of D; for otherwise L would divide D
into a quadrilateral and a triangle, not two triangles. For the two triangles to
be congruent, L must bisect the angle at its vertex, forming two equal right
angles. Also L must be perpendicular to the side opposite that vertex.  Also L
is a common side of the two triangles. Thus D is isosceles and L divides D into
two right triangles.

We check the choices individually to see if they are true:

(a) If D is isosceles, then D is equilateral.

That is not necessarily true, for D's vertex angle could be 90<sup>o</sup> and the base 
angles could be 45<sup>o</sup> each. 

(b) If D is right, then D is equilateral.

That could never be true for eq1uilateral triangles have only three 60<sup>o</sup>
interior angles and no right angles. 

(c) L is parallel to a side of D.

That could not be true for then L would divide D into a triangle and a
quadrilateral, not two triangles. 

(d) L is longer than the distance from either endpoint of L to the centroid
    of D.

This is true because L is the median of D drawn from the apex of isosceles
triangle D. The centroid of a triangle is 2/3 of the distance from a vertex
to the midpoint of the opposite side.  The centroid is a point along L, and not
an endpoint of L, thus its distance from either endpoint is less than the length
of L. 

Answer: (d)

Edwin</pre>