Since we know that #24# is the longest side, we can assume that is is similar to the longest side of the first triangle. Already, we know that we have a scale factor #24:15 or 24/15#. If we simplify that down, we get #8/5#.
Now we can scale the other 2 sides using this scale factor. Since our scale is #"large"/"small"# we need to set up a proportion in the same way. That means that our proportions will look like #8/5 = x/11# and #8/5 = x/5#. When we cross multiply we get #88 = 5x# and #40=5x#. We can solve them out to get that our side lengths are #8 and 17.6#.
We can add all 3 sides together, #24 + 17.6 + 8 = P#, to get #P = 49.6#.
To check your answer, add all sides of the first triangle together. That equals #31#. Now do #49.6/31 = 8/5# (since our scale factor should come out to be #8/5#, like the rest of the triangle). It should be the same. #248 = 248#, so our answer is correct.
