Question 726768
If the lengths of the sides of triangle ABC are 3, 6, and 4 (in whatever units),
ABC has a perimeter of {{{3+6+4=13}}} (measured in the same units).
If the angles of triangle XYZ have the same measures as the angles in ABC,
the triangles are similar, meaning that one is a scaled-up version of the other.
If the length of one side of XYZ is 24, XYZ is a larger, scaled-up version of ABC.
We just do not know the scale-up factor, but there are three possibilities.
 
If the side measuring 24 is the longest, corresponding the the side measuring 6 in ABC,
the scale-up factor is {{{24/6=4}}} and everything in XYZ is 4 times longer than in ABC, including the perimeter.
That means the perimeter of XYZ is
{{{4*13=52}}}
 
If the side measuring 24 is the medium length one,=, corresponding the the side measuring 4 in ABC,
the scale-up factor is {{{24/4=6}}} and everything in XYZ is 6 times longer than in ABC, including the perimeter.
That means the perimeter of XYZ is
{{{6*13=78}}}
 
If the side measuring 24 is the shortest, corresponding the the side measuring 3 in ABC,
the scale-up factor is {{{24/3=8}}} and everything in XYZ is 8 times longer than in ABC, including the perimeter.
That means the perimeter of XYZ is
{{{8*13=104}}}
 
SO the perimeter of XYZ could be {{{highlight(54)}}}, or {{{highlight(78)}}}, or {{{highlight(104)}}}