You plan to paint the outside of this barn, including the roof, but not the floor. Which calculation will help you best decide how much paint you will need?

Lateral surface area of the barn

Step-by-step explanation:

In this case, the area will include the walls making the barn and the roof. It will exclude the floor that is not painted.The area of the walls can be calculated by applying the formula for area of a rectangle or area of a square depending on the dimensions of of the walls. The roof will include area for the rectangular parts and the triangular parts which also will depend on the shape of the roof.

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.