What does it mean for shapes to be mathematically similar?
basically for pictures to be mathematically similar they have to be perfect enlargements or reductions of another picture. In other words you can have a triangle with all sides of 2 and a triangle with all sides of 6 and they would be similar because if we divide 6 by 3 than we can turn our image triangle into a copie of our original. But, if the second triangle had 2 sides of 6 and one side of 5.5 even tho the triangle look much alike they are no longer mathematically similar.
Scale factor
what is scale factor?
scale factor is a number we use to multiply the original by to fabricate an image.
so if we have a square with the sides of 5cm and we want to make it twice bigger we are going to have a scale factor of 2 and the formula to make an image is (Original x Scale factor = Image)
so we replace make the formula:
5 x 2 = x
x = 10
meaning that the square will have sides of 10cm long.
In some scenarios like if you know the original’s side A is 3 and the Image’s side A is 9 you can use this formula to find the scale factor
Original x Scale factor = Image
3x = 9
x = 3
3 = scale factor
And by finding the scale factor of 1 side we now know how long all of the sides are if we know 1 of the measurements from that side.
lets say we know the scale factor from Original to Image is 5 than we also know the scale factor of Image to Original because it’s the reciprocal of the Original to Image which is 
Enlargements/reductions
We can tell if a shape is getting an enlargement or reduction by looking at its scale factor. If the scale factor is bigger than 1 than it is getting an enlargement otherwise if its smaller than 1 than its a reduction.
As we can see this is a 2 cm sided square on a grid to make an enlargement of double we would have to count out how long each side is and draw them twice bigger like this:
On a graph enlargements look a bit different but still are enlargements of the original. For making enlargements on a graph you need to start with one of the points of the original for example the point is 3,7 and you want to make the image 3 times bigger (aka a scale factor of 3) than what you want to do is multiply both of the points by 3 so your new points are 9,21 and you repeat the process with all of the other points and your image(s) should look something like this.
How to mesure thing indirectly using shadows or a mirror
using shadows or a mirror you can mesure objects!
so to start if you have 2 objects and you look at their shadows at the same time and you know the size of one of them than you can figure out the size of the other one!
Ex: if you are 5.5 f tall and your shadow mesures 16.5 f and you have a nearby tree with a shadow of 48 f than all you have to do is find the scale factor for the shadow to the original. With that scale factor you times it with the 48 f from the tree and you will find the actual height of the tree!
my work:
16.5x = 5.5
x =
(so the scale factor is 
15
So in the end we found out that the tree was 15 f long.