Perimeter of Triangle in a Right triangle. I am having difficulty solving this problem:


The perimeter of a right triangle is 18 inches. If the midpoints of three sides are joined by line segments they form another triangle . What is the perimeter of this new triangle ? (Ans: 9 inches) . 


Any suggestions on how to solve it ?

Each side of the new triangle is half the side of the old triangle, meaning that the perimeter is halved as well.
For a little more detail, draw it out. cutting at the midpoint makes this new triangle one-half of a rectangle. So clearly, since you cut at the midpoint, the legs of the new triangle are half the legs of the old, since opposite sides are of equal length in a rectangle. By similar triangles the hypotenuse is halved as well. So the whole perimeter is cut in half and so the answer is $9$.
EDIT: Picture! The rectangle I'm talking about is $AGFE$.

$AE$ is half of the side, so $GF$ is also half of that side. The same argument can be made for $FE$ and $GE$ easily. This argument doesn't need a right angle though: what matters is that you form a parallelogram (opposite sides are of the same length), which always happens. This is just a special case.
