document.write( "Question 1210463: A straight line \ell divides a triangle \Delta into two congruent triangles. Select all the statements that must be true.\r
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\n" ); document.write( "\n" ); document.write( "(a) If Delta is isoceles, then it is equilateral\r
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Algebra.Com's Answer #852774 by mccravyedwin(421)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I suspect that this problem was a mistranslation into English, as it could have\r\n" );
document.write( "been mistranslated from a language where \"centroid\" could have translated as\r\n" );
document.write( "\"midpoint\". Unlike tutor Ikleyn, I never write anything assuming the student did\r\n" );
document.write( "anything wrong besides mistyping or mistranslating into English.  I think\r\n" );
document.write( "perhaps the problem should have been stated this way:\r\n" );
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document.write( "A straight line segment L divides triangle D into two congruent triangles.\r\n" );
document.write( "Select all the statements that must be true.\r\n" );
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document.write( "(a) If D is isosceles, then D is equilateral.\r\n" );
document.write( "(b) If D is right, then D is equilateral.\r\n" );
document.write( "(c) L is parallel to a side of D.\r\n" );
document.write( "(d) L is longer than the distance from either endpoint of L to the centroid\r\n" );
document.write( "    of D.\r\n" );
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document.write( "A straight line segment L divides triangle D into two congruent triangles. \r\n" );
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document.write( "This means one end of L must be at a vertex of D; for otherwise L would divide D\r\n" );
document.write( "into a quadrilateral and a triangle, not two triangles. For the two triangles to\r\n" );
document.write( "be congruent, L must bisect the angle at its vertex, forming two equal right\r\n" );
document.write( "angles. Also L must be perpendicular to the side opposite that vertex.  Also L\r\n" );
document.write( "is a common side of the two triangles. Thus D is isosceles and L divides D into\r\n" );
document.write( "two right triangles.\r\n" );
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document.write( "We check the choices individually to see if they are true:\r\n" );
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document.write( "(a) If D is isosceles, then D is equilateral.\r\n" );
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document.write( "That is not necessarily true, for D's vertex angle could be 90o and the base \r\n" );
document.write( "angles could be 45o each. \r\n" );
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document.write( "(b) If D is right, then D is equilateral.\r\n" );
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document.write( "That could never be true for eq1uilateral triangles have only three 60o\r\n" );
document.write( "interior angles and no right angles. \r\n" );
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document.write( "(c) L is parallel to a side of D.\r\n" );
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document.write( "That could not be true for then L would divide D into a triangle and a\r\n" );
document.write( "quadrilateral, not two triangles. \r\n" );
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document.write( "(d) L is longer than the distance from either endpoint of L to the centroid\r\n" );
document.write( "    of D.\r\n" );
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document.write( "This is true because L is the median of D drawn from the apex of isosceles\r\n" );
document.write( "triangle D. The centroid of a triangle is 2/3 of the distance from a vertex\r\n" );
document.write( "to the midpoint of the opposite side.  The centroid is a point along L, and not\r\n" );
document.write( "an endpoint of L, thus its distance from either endpoint is less than the length\r\n" );
document.write( "of L. \r\n" );
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document.write( "Answer: (d)\r\n" );
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document.write( "Edwin
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