Similarity of rectangles
Screens, photos and sheets of paper are rectangles, and you hardly ever think about whether they are "similar" to one another. But when you enlarge a photo, choose a screen resolution or photocopy something at a different size, you are using (or breaking) the similarity of rectangles without realising it.
When two rectangles are similar
Check whether a tablet screen with a resolution of $2048 \times 1536$ pixels is mathematically similar to a computer monitor screen with a resolution of…
Check whether a tablet screen with a resolution of $2048 \times 1536$ pixels is mathematically similar to a computer monitor screen with a resolution of $2560 \times 1440$ pixels. Justify your answer by calculating the ratio of scale factor between their sides.
A printed photograph has dimensions of $10 \text{ cm}$ wide by $15 \text{ cm}$ long. We want to enlarge it for a poster so that it is similar to the original. If the…
A printed photograph has dimensions of $10 \text{ cm}$ wide by $15 \text{ cm}$ long. We want to enlarge it for a poster so that it is similar to the original. If the width of the enlargement must be $60 \text{ cm}$, how long will it be?
Screen formats: the aspect ratio
In video and photography, the "aspect ratio" (for example $16{:}9$) is precisely the base/height ratio that defines a whole family of similar rectangles, whether it is a mobile phone screen or a cinema screen.
The aspect ratio of a modern widescreen television is $16:9$, which means that the rectangle of its screen is similar to a rectangle with a base of $16 \text{ cm}$ and a…
The aspect ratio of a modern widescreen television is $16:9$, which means that the rectangle of its screen is similar to a rectangle with a base of $16 \text{ cm}$ and a height of $9 \text{ cm}$. If the base of a television screen measures $104 \text{ cm}$, what is its height?
A television screen has a diagonal of $55 \text{ pulgadas}$ and its format is similar to a $16 \times 9$ rectangle. Knowing that $1 \text{ pulgada}$ is equivalent to…
A television screen has a diagonal of $55 \text{ pulgadas}$ and its format is similar to a $16 \times 9$ rectangle. Knowing that $1 \text{ pulgada}$ is equivalent to $2,54 \text{ cm}$, calculate the exact dimensions (base and height) of the screen in centimetres.
The DIN format: folding in half without losing the proportion
The DIN paper sizes (A4, A3, A5...) have a special property: when you cut a sheet in half, parallel to the short side, the two pieces are similar to the original sheet.
A standard sheet of A4 paper measures $210 \text{ mm}$ wide by $297 \text{ mm}$ long. If we cut it in half with a cut parallel to the shorter side, we obtain two A5…
A standard sheet of A4 paper measures $210 \text{ mm}$ wide by $297 \text{ mm}$ long. If we cut it in half with a cut parallel to the shorter side, we obtain two A5 sheets. Calculate the dimensions of the A5 paper and divide its longer side by its shorter side to check whether it maintains approximately the same ratio as the original A4 paper.
The DIN paper format is based on the geometric property that, when a sheet is cut in half parallel to its shorter side, the two new rectangles obtained are similar to…
The DIN paper format is based on the geometric property that, when a sheet is cut in half parallel to its shorter side, the two new rectangles obtained are similar to the original rectangle. Prove algebraically that for a rectangle with sides $a$ and $b$ (with $a > b$) to satisfy this property, the ratio of the longer side to the shorter side must be exactly $\sqrt{2}$.
Frames and enlargements: when similarity is *not* preserved
Adding a frame of constant thickness to a rectangle, or a strip of fixed width, almost never preserves similarity: the thickness has to be distributed carefully for it to do so.
A rectangular wooden sheet of $120 \text{ cm}$ length and $90 \text{ cm}$ width is available. A single straight cut parallel to the shorter side is to be made to discard…
A rectangular wooden sheet of $120 \text{ cm}$ length and $90 \text{ cm}$ width is available. A single straight cut parallel to the shorter side is to be made to discard a strip of width $x$, such that the remaining wooden rectangle is similar to the original rectangle. Calculate the value of $x$.
A rectangular picture has dimensions of $a$ (base) and $b$ (height), with $a > b$. A wooden frame is added, which has a horizontal thickness $x$ (on the left and right…
A rectangular picture has dimensions of $a$ (base) and $b$ (height), with $a > b$. A wooden frame is added, which has a horizontal thickness $x$ (on the left and right sides) and a vertical thickness $y$ (at the top and bottom). Prove algebraically that, for the outer rectangle of the frame to be similar to the inner rectangle of the picture, the thicknesses must strictly satisfy the ratio $\frac{x}{y} = \frac{a}{b}$.