Since triangles ABC and TUV are similar, their corresponding sides are proportional. We can use this fact to find the missing side lengths of triangle TUV.
First, we need to find the scale factor between the two triangles. We can do this by comparing the lengths of corresponding sides.
The longest side of triangle ABC is 50, and the longest side of triangle TUV is 275. Therefore, the scale factor is:
275/50 = 5.5
This means that all corresponding sides of triangle TUV are 5.5 times larger than the corresponding sides of triangle ABC.
To find the perimeter of TUV, we need to add up the lengths of all three sides. We know that the longest side is 5.5 times larger than the longest side of ABC, which is 50.
Therefore, the length of the longest side of TUV is:
50 x 5.5 = 275
The other two sides of TUV are also 5.5 times larger than their corresponding sides in ABC. These sides have lengths of:
40 x 5.5 = 220
24 x 5.5 = 132
To find the perimeter, we add up all three sides:
275 + 220 + 132 = 627
Therefore, the perimeter of TUV is 627.