If i'm given the lengths of two similar rectangles but not the width, such as lengths x^2 and xy, how do you go about finding the ratio of thier areas?
Similar figures have the same "shape", but not necessarily the same size. For polygons in particular, the corresponding angles are congruent and corresponding side lengths are proportional. The constant of proportionality is called the scale factor. (This is also the dilation factor as similar figures can be considered as dilations of a preimage to an image.)
If the scale factor is a:b then all corresponding lengths are in the ratio of a:b. (e.g. lengths of corresponding sides, lengths of corresponding diagonals, radii, apothems, chords, etc...)
If the scale factor is a:b then the ratio of corresponding areas is a^2:b^2. (e.g. area, lateral area, etc...)
** For polyhedra, if the scale factor is a:b corresponding volumes are in the ratio a^3:b^3. **
We are given similar rectangles whose lengths are x^2 and xy. Since the rectangles are similar, all corresponding lengths are proportional and the scale factor is x:y. (Write the proportion x^2:xy in simplest form; this is the scale factor.)
All corresponding lengths, including perimeters, are in the same proportion. All corresponding areas are in the ratio x^2:y^2.
The ratio of their areas is x^2:y^2.
Let the length and breadth of two similar rectangles be `l_1,b_1` and `l_2,b_2` respectively.
Since the rectangles are similar,
Area of first rectangle `A_1=l_1*b_1`
Area of second rectangle `A_2=l_2*b_2`
Now we are given, `l_1=x^2, l_2=xy`