Exercises: Similarity of rectangles

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 width of the enlargement must be $60 \text{ cm}$, how long will it be?

A lawn plot is rectangular and measures $40 \text{ m}$ wide by $60 \text{ m}$ long. On the architectural plan, the width of the plot is drawn with a length of $8 \text{ cm}$. What is the area occupied by the plot on the plan expressed in $\text{cm}^2$?

Two rectangles are similar. The dimensions of the first rectangle are $5 \text{ cm}$ and $12 \text{ cm}$. Given that the perimeter of the second rectangle is $102 \text{ cm}$, determine the lengths of its sides.

The canvas of a painting has a base of $40 \text{ cm}$ and a height of $60 \text{ cm}$. A passe-partout (cardboard mount) $10 \text{ cm}$ thick is added along each of its four sides. Is the outer rectangle of the mount similar to the inner rectangle of the canvas? Justify your answer by calculating the ratios.

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?

Rectangle $A$ has dimensions of $x$ cm in height and $18 \text{ cm}$ in base. Rectangle $B$ measures $12 \text{ cm}$ in height and $27 \text{ cm}$ in base. Knowing that both rectangles are similar and that the base is the longer side in both cases, calculate the exact value of $x$.

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.

Mateo wants to enlarge an ID photo measuring $3\text{ cm}$ in base by $4\text{ cm}$ in height to place it in a picture frame. To ensure the image is not distorted, he reasons as follows: “If I add $5\text{ cm}$ to the $3\text{ cm}$ base, it becomes $8\text{ cm}$; therefore, to maintain the shape I must add exactly the same to the height: $4 + 5 = 9\text{ cm}$. Since I added the same length to both sides and all angles remain right angles ($90^\circ$), the new $8 \times 9\text{ cm}$ rectangle is similar to the original photo.”\n\na) Identify the conceptual mistake Mateo makes regarding the similarity of rectangles.\nb) Numerically justify why the $8 \times 9\text{ cm}$ rectangle is not similar to the $3 \times 4\text{ cm}$ one by calculating and comparing their aspect ratios.\nc) Correct Mateo's calculation: if the enlarged base must be $8\text{ cm}$, what must the height be for the photograph to maintain strict geometric similarity with the original?

The regulations of the Royal Spanish Football Federation (RFEF) allow some flexibility in the pitch dimensions for official competitions. The pitch at the Estadio de Vallecas (Madrid) measures approximately $100\text{ m}$ in length by $65\text{ m}$ in width, while the one at the Estadio de Mestalla (Valencia) measures $105\text{ m}$ in length by $68\text{ m}$ in width.\n\na) Before performing any numerical calculation, estimate mentally whether you think both pitches are similar or if one of them seems noticeably more elongated than the other. Justify your initial intuition in one sentence.\nb) Calculate the ratio of length to width (aspect ratio) for both pitches, rounding to two decimal places.\nc) Check whether your initial estimate was correct, formally determine whether the two pitches are mathematically similar, and assess whether a footballer would notice a significant difference in shape in practice or if the proportion is almost identical.

A rectangular window measures $80 \text{ cm}$ wide and $120 \text{ cm}$ high. An aluminium frame of constant thickness $x$ is to be installed around the entire window. Formulate an equation and show whether there is any value of $x > 0$ for which the outer rectangle of the frame is mathematically similar to the inner rectangle of the window.

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.

An identity card has dimensions of $85,6 \text{ mm}$ by $53,98 \text{ mm}$. A golden rectangle is one in which the ratio of its longer side to its shorter side is the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$.

1. a) Check whether the identity card is an approximate golden rectangle by calculating its ratio.
2. b) If the width of the card were exactly $54 \text{ mm}$, what should its exact length be, rounded to two decimal places, to be a perfect golden rectangle?

A professional football pitch is rectangular with dimensions of $105 \text{ m}$ long and $68 \text{ m}$ wide. For tactical practice, the coach draws a similar pitch on the whiteboard with an area of $1785 \text{ cm}^2$. What are the dimensions in centimetres of the pitch drawn on the whiteboard?

You can do it on paper and send a photo with your answer.

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 $x$.

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 $\sqrt{2}$.

A rectangle is inscribed inside an isosceles triangle of base $10 \text{ cm}$ and height $6 \text{ cm}$ such that one of its sides lies on the base of the triangle and the other two vertices touch the equal sides of the triangle. Knowing that the inscribed rectangle has a $2:1$ format (its base is twice its height), calculate its exact dimensions.

The vertices of a rectangle in the Cartesian plane are $A(1, 1)$, $B(4, 1)$, $C(4, 3)$ and $D(1, 3)$. An enlargement with centre at the origin $O(0, 0)$ and scale factor $k = -2$ is applied to obtain a new rectangle $A'B'C'D'$.

1. a) Calculate the coordinates of the new vertices.
2. b) Calculate the area of the original rectangle and the transformed rectangle, and verify the relationship between both areas using the square of the scale factor.

In Spain, the National Identity Document (DNI) and bank cards follow the international ID-1 standard (ISO/IEC 7810 standard), whose official measurements are $85{,}60\text{ mm}$ in length by $53{,}98\text{ mm}$ in width. On the other hand, official public administration forms are published on DIN A4 paper sheets, measuring $297\text{ mm}$ in length by $210\text{ mm}$ in width.\n\na) Model the situation by calculating the aspect ratio (length divided by width) of both formats, rounding to three decimal places. Based on the results, argue whether the DNI is similar to a DIN A4 sheet.\nb) Sheets in the DIN series satisfy the condition that their exact theoretical ratio is $\sqrt{2} \approx 1{,}414$. If the Fábrica Nacional de Moneda y Timbre decided to redesign the card so that it kept its current width of exactly $53{,}98\text{ mm}$ but was strictly similar to a DIN A4 sheet, what should its new length be?\nc) Compare the actual length of the DNI ($85{,}60\text{ mm}$) with the one you obtained in section b). Is the real DNI more elongated or squarer than the DIN format? Justify your answer by comparing both ratios.

A digital advertising agency designs responsive banners for online media. They have created a main ad for desktop screens measuring $960\text{ pixels}$ wide by $360\text{ pixels}$ high. For the mobile version, they need the new ad to be mathematically similar to the desktop one so that logos are not distorted or cropped.\n\na) Write the equation that must relate the base $x$ and the height $y$ of the mobile ad to ensure similarity with the desktop one, expressing the constant of proportionality as an irreducible fraction.\nb) If the space on the phone screen requires the ad height to be exactly $120\text{ pixels}$, determine what width $x$ the banner must have.\nc) Set up and solve the inverse problem involving area: if the client requires the mobile ad to have an exact total area of $15000\text{ pixels}^2$ while preserving similarity with the desktop version, calculate what its exact dimensions (base and height) must be.

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 $\frac{x}{y} = \frac{a}{b}$.

A rectangle of dimensions $L \times x$ (where $x < L$) is attached to a square of side $L$, sharing a side of length $L$. The total rectangle formed, of dimensions $L \times (L+x)$, is required to be mathematically similar to the small attached rectangle.

1. a) Set up the equation of similarity and calculate the exact value of $x$ in terms of $L$.
2. b) Show that the total rectangle is a golden rectangle.

The "silver ratio" is defined in mathematics as $\delta_S = 1 + \sqrt{2}$. A silver rectangle is one in which the ratio of its longer side to its shorter side is exactly $\delta_S$. Prove algebraically that if two maximal squares (whose sides are equal to the height of the rectangle) are cut from a silver rectangle, the remaining rectangle is still a silver rectangle.

In a coordinate system, the profile of an asymmetrical pitched roof is bounded by the x-axis and the line segments joining the origin $(0,0)$ to the point $(4,6)$, and the latter to the point $(12,0)$. A rectangular solar panel is to be placed inscribed in this triangle, resting on the $X$ axis. If the solar panel is required to be similar to the rectangle circumscribing the whole triangle (that is, a rectangle of $12 \times 6$), calculate the exact dimensions of the solar panel.

Consider a rectangle $R_1$ with dimensions $21 \times 13 \text{ cm}$. The largest possible square is cut from it, giving a new rectangle $R_2$. From this $R_2$, the largest possible square is cut again to obtain $R_3$, and so on. Calculate the dimensions of $R_2$, $R_3$ and $R_4$. Is any of these rectangles strictly similar to $R_1$? Calculate the ratio of the longer side to the shorter side at each step and explain which irrational value this sequence of rectangles approaches.

A rectangular solar farm with dimensions $a$ (length) and $b$ (width), with $a > b$, is to be subdivided into $n$ identical rectangular plots by cuts parallel to its shorter side. If each of the $n$ plots is required to be mathematically similar to the original solar farm, determine the algebraic expression that the ratio $\frac{a}{b}$ must satisfy in terms of the number of plots $n$.

Two rectangles $R_1$ and $R_2$ are similar with a scale factor $k > 1$ (where $R_2$ is an enlargement of $R_1$). We know that the diagonal of $R_1$ measures $D$ and that the perimeter of $R_1$ is $P$. Find an algebraic expression to calculate the area of the enlarged rectangle $R_2$ in terms only of $k$, $P$ and $D$.

Consider a rectangle $ABCD$ with base $AB = 8 \text{ cm}$ and height $BC = 6 \text{ cm}$. Draw the diagonal $AC$. From the vertex $B$, draw the perpendicular to the diagonal, intersecting it at point $H$. The segment $BH$ is the altitude of the right-angled triangle $ABC$ to the hypotenuse. Prove, by calculating their exact dimensions, that the rectangle whose sides are the segments $AH$ and $BH$ is mathematically similar to the original rectangle $ABCD$.

You can do it on paper and send a photo with your answer.

In an editorial design workshop, two rectangular prints need to be compared to check whether their layouts are similar:\n- Print 1: base of $45\text{ cm}$ and height of $60\text{ cm}$.\n- Print 2: base of $54\text{ cm}$ and height of $72\text{ cm}$.\n\nThe team debates which method to apply, and three proposals emerge:\n- Method A (Internal aspect ratio): Divide the height by the base within each print ($60/45$ and $72/54$) and see if both quotients match.\n- Method B (Scale factor or external ratio): Divide corresponding sides between both prints ($54/45$ and $72/60$) to find the enlargement factor $k$.\n- Method C (Cross-multiplication): Set up the equality $\frac{45}{60} = \frac{54}{72}$ and check whether the product of the extremes equals the product of the means ($45 \cdot 72 = 60 \cdot 54$).\n\na) Apply all three methods using the data from the statement and confirm whether the prints are similar.\nb) Compare the methods and justify: which one directly tells you how many times larger one print is than the other, which one describes the inherent shape of the print independently of the other, and which one is the safest if the measurements produce repeating non-terminating decimals?\nc) If you are asked to calculate the height $H$ of a third similar print whose base measures $75\text{ cm}$, choose with justification which of the three methods allows you to solve for $H$ most directly, and find its value.

A professional photographer takes pictures with a digital SLR camera whose sensor produces images with the standard photography ratio of $3:2$ (base greater than height). When preparing an exhibition, he analyzes the printing and framing options available in the Spanish market:\n\na) At the photo lab, he is offered two standard paper sizes: $10 \times 15\text{ cm}$ and $13 \times 18\text{ cm}$. Model and analytically prove which of the two papers is strictly similar to the sensor image and which will cause distortion or cropping if printed filling the entire paper.\nb) For the $13 \times 18\text{ cm}$ paper, if the photographer fits the photo to the full width of $18\text{ cm}$ while keeping the $3:2$ ratio of the original image intact, what actual height will the printed image occupy? How many centimeters of paper will remain blank along the height?\nc) To an already printed $10 \times 15\text{ cm}$ photo, a frame with a white passe-partout (mat) of uniform constant thickness $x = 2{,}5\text{ cm}$ on all four sides is added. Prove algebraically whether the overall outer rectangle ($15 \times 20\text{ cm}$) is similar to the original $10 \times 15\text{ cm}$ photo or not. From the theoretical condition $\frac{x}{y} = \frac{a}{b}$, calculate what vertical thickness $y$ the frame should have if the horizontal thickness is set to $x = 2{,}5\text{ cm}$ so that the outer rectangle is indeed similar to the photograph.