BD is a perpendicular bisector of triangle ABC. XZ is a perpendicular bisector of triangle WXY.

Triangle ABC is similar to triangle WXY with a scale factor of 1:3.
Part I. Find BD, AB, WZ, XZ and WX.
Part II. What is the scale factor of the perimeters of triangle ABC and triangle WXY?

Part I:

To find the lengths of BD, AB, WZ, XZ, and WX, we need to use the given information and the properties of perpendicular bisectors and similar triangles.

1. BD - Since BD is a perpendicular bisector of triangle ABC, it intersects AC at its midpoint (let's call it M). As a perpendicular bisector, BD also passes through the circumcenter of triangle ABC, which is equidistant from all three vertices. Therefore, BD is equal to the radius of the circumcircle. In a circumcenter, the radius is the distance from the center to any of the triangle's vertices. Hence, BD is equal to the radius of the circumcircle of triangle ABC.

2. AB - Since triangle ABC is similar to triangle WXY with a scale factor of 1:3, we can use the concept of corresponding sides to find AB. The corresponding side of AB in triangle WXY would be WX. So, AB is one-third of WX.

3. WZ - Similar to BD, WZ is a perpendicular bisector of triangle WXY. It intersects XY at its midpoint (let's call it N). Therefore, WZ is equal to the radius of the circumcircle of triangle WXY.

4. XZ - Similar to WZ, XZ is also equal to the radius of the circumcircle of triangle WXY.

5. WX - WX is the side of triangle WXY. Since triangle ABC is similar to triangle WXY with a scale factor of 1:3, we know that corresponding sides are proportional. Hence, WX is three times the length of AB.

Part II:

To find the scale factor of the perimeters of triangle ABC and triangle WXY, we need to calculate the ratios of their corresponding sides.

The ratio of corresponding sides is equal to the scale factor. In this case, since triangle ABC is similar to triangle WXY with a scale factor of 1:3, the scale factor is 1:3.

Therefore, the scale factor of the perimeters of triangle ABC and triangle WXY is also 1:3.