Exercises: Similarity

A graphic designer is working on the logo for a new app. The original design is a rectangle measuring $8\text{ cm}$ wide and $12\text{ cm}$ high. For the loading screen, he needs to create a larger version that is similar to the original. If the width of the enlarged version must be $20\text{ cm}$, what will be the height of the logo on that screen?

On the floor plan of a house, the scale used is $1:50$. If the living room has a length of $12\text{ cm}$ in the drawing, calculate the actual length of this room in metres.

Two triangles are positioned in Thales's configuration. Knowing that the horizontal segments are parallel and measure $6\text{ cm}$ and $10\text{ cm}$, and that the vertical segment of the smaller triangle measures $9\text{ cm}$, calculate the value of the unknown vertical segment denoted by $x$ of the larger triangle.

A 3rd Year student is $1.60\text{ m}$ tall. At a certain time of day, they cast a shadow on the ground of $2\text{ m}$. If right next to them, a lamppost casts a shadow of $5\text{ m}$, what is the height of the lamppost?

State whether the following pairs of triangles are similar and justify which criterion you used (AAA, SSS or SAS):

• Triangle 1: Angles of $40^\circ$ and $70^\circ$.

• Triangle 2: Angles of $70^\circ$ and $70^\circ$.

A pizzeria offers two sizes of round pizza that are similar. The "Small" pizza has a radius of $10\text{ cm}$ and an area of approximately $314\text{ cm}^2$. If the "Family" pizza has a radius of $20\text{ cm}$, what will its area be? (Remember the relationship between the scale factor $k$ and the areas).

Cian wants to make an enlarged copy of a rectangular art print measuring $15\text{ cm} \times 20\text{ cm}$. At the print shop, they offer to enlarge it proportionally so that the new width is $45\text{ cm}$. Cian calculates mentally: “Since the width goes from $15\text{ cm}$ to $45\text{ cm}$, the figure is tripled. The original print cost $€2$, so the new print, having three times the size, will cost me exactly $€6$.”

1. a) Identify the error in Cian's reasoning about the cost of the paper. Which geometric quantity determines the material expense: linear length or area?
2. b) Find the linear scale factor $k$ and calculate the dimensions (width and length) and the area of the enlarged print.
3. c) If the price of the special printing paper is proportional to the printed surface area, what is the fair and actual price that Cian should pay for the large print?

In a town square in Segovia, at eleven in the morning, a vertical traffic sign $2\text{ m}$ high casts a shadow of $1.6\text{ m}$ on the pavement. At that exact same moment, the tower of a nearby church casts a straight shadow of $28\text{ m}$.

1. a) Before calculating with precision, make a reasoned estimate: observing the relationship between the height of the sign and its shadow, do you expect the height of the tower to be greater or smaller than the length of its shadow? Will it be closer to $20\text{ m}$, $35\text{ m}$, or $50\text{ m}$?
2. b) Set up the corresponding mathematical proportion (using similar triangles formed by the sun's rays) and calculate the height of the tower exactly.
3. c) Compare the numerical value obtained with your previous estimate and evaluate whether the result makes physical and architectural sense for a historic bell tower.

A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with side lengths $6\text{ cm}$, $8\text{ cm}$, $9\text{ cm}$ and $12\text{ cm}$. The client asks for a larger version for a billboard, where the side that previously measured $9\text{ cm}$ must now measure $45\text{ cm}$.

1. a) Calculate the scale factor $k$ between the billboard logo and the original.
2. b) Determine the lengths of the other three sides in the billboard logo.
3. c) If the perimetre of the original logo is $35\text{ cm}$, what will be the perimetre of the billboard logo without adding its sides one by one?

To calculate the height of a communication tower that casts a very long shadow, Seán, who is $1.70\text{ m}$ tall, stands so that the tip of his shadow coincides with the tip of the tower's shadow. If Seán is $25\text{ m}$ from the base of the tower and his shadow measures $2.5\text{ m}$, answer:

1. a) Why can we state that the triangles formed by Seán and the tower with their respective shadows are similar? Mention the similarity criterion used.
2. b) Calculate the height of the tower.

In the following figure, the lines $r$, $s$, and $t$ are parallel. Knowing that $y = x + 1$, use Thales's Theorem to calculate the unknown lengths $x$ and $y$ expressed in centimetres.

A cinema projector is located $6\text{ metres}$ from a screen. The beam of light projects an image that has a height of $2.4\text{ metres}$ on the screen.

1. a) If we place an obstacle of height $40\text{ cm}$ parallel to the screen to create a shadow play, at what distance from the projector should we place it so that its shadow covers the entire height of the screen?
2. b) If we move the projector further back to $9\text{ metres}$ from the screen, what will be the new height of the projected image?

Two rectangles are similar. The first has a base of $12\text{ cm}$ and an area of $96\text{ cm}^2$. The second rectangle has an area of $150\text{ cm}^2$.

1. a) Calculate the scale factor between the second and first rectangle. (Hint: use the relationship between areas).
2. b) Calculate the dimensions (base and height) of the second rectangle.
3. c) What is the ratio between their perimetres?

Observe the right-angled triangle in the figure. The altitude $h$ to the hypotenuse divides the original triangle into two smaller triangles.

1. a) Justify why the three triangles (the original and the two small ones) are similar to each other.
2. b) If the segments into which the hypotenuse is divided measure $m = 4\text{ cm}$ and $n = 9\text{ cm}$, calculate the altitude $h$ using the similarity of the two small triangles.
3. c) Calculate the area of the original triangle.

A furniture company manufactures regular hexagonal coffee tables. They have two models: the "Mini" model has a side length of $30\text{ cm}$ and its glass top costs $\text{€}\ 45$. The "Gran Salón" model is similar to the previous one and has a side length of $60\text{ cm}$.

1. a) What is the scale factor between the large model and the small one?
2. b) Knowing that the price of the glass is proportional to its area, how much will the glass top of the "Gran Salón" model cost?
3. c) If $180\text{ cm}$ of metal tape is needed to border the small table, how much tape will be needed for the large table?

On an Ordnance Survey Ireland (OSi) map with a scale of $1:50\,000$, a forest firefighter crew working with Coillte analyzes a triangular plot of woodland. On the paper, the three sides of the triangle measure exactly $3\text{ cm}$, $4\text{ cm}$, and $5\text{ cm}$.

1. a) Verify numerically that the triangle on the map is a right triangle and calculate its perimeter (in $\text{cm}$) and its area (in $\text{cm}^2$).
2. b) Using the scale as the linear scale factor, calculate the real lengths of the three sides of the forest land in kilometres and find its real perimeter.
3. c) Calculate the real surface area of the plot in two independent ways: first, by calculating the area from the real lengths in kilometres from the previous part; second, by multiplying the map area by the appropriate area ratio $k^2$. Verify that both methods yield the same result and express the final result in hectares ($1\text{ km}^2 = 100\text{ ha}$).

An artisanal ceramics workshop in Kilkenny manufactures similar decorative parallelogram-shaped tiles. The basic mould has a base of $12\text{ cm}$, a height of $5\text{ cm}$, and slanted lateral sides of $6.5\text{ cm}$. For a mural in a public building, they receive an order to manufacture similar pieces whose individual surface area is exactly $1\,500\text{ cm}^2$.

1. a) Calculate the area and perimeter of the basic tile.
2. b) Set up the mathematical relationship between the areas of two similar figures to deduce the linear scale factor $k$ that transforms the basic tile into the large mural tile.
3. c) Using this ratio $k$, calculate the full dimensions (base, height, and oblique sides) of the large tile, as well as the metres of protective tape that will be needed to wrap around the perimeter of each of the large pieces.

An architect is designing the facade of a modern auditorium. He has drawn a main right-angled triangle and, to improve acoustics, needs to draw the altitude to the hypotenuse. This altitude divides the original triangle into two smaller triangles. Knowing that the legs of the original triangle measure $18\text{ m}$ and $24\text{ m}$, use triangle similarity to calculate:

1. a) The length of the altitude drawn to the hypotenuse.
2. b) The lengths of the two segments into which the hypotenuse is divided.

A 3D printing company has created a scale model of a circular square. In the model, the area of the square is $450\text{ cm}^2$. The client requests a second model where the perimetre is $20\%$ larger than that of the first model.

1. a) What will be the area of the square in this second model?
2. b) If the actual square has a radius of $30\text{ m}$, what scale was used in the first model? (Express the scale in the format $1:n$).

To measure the height of a cooling tower at a power station, a team of engineers uses a mirror placed on the ground. The engineer stands such that, looking into the mirror, he sees the top of the tower reflected. The mirror is $45\text{ m}$ from the base of the tower. The engineer, whose eyes are at a height of $1.75\text{ m}$, is $1.5\text{ m}$ from the mirror. Between the mirror and the tower there is a safety fence of height $5\text{ m}$.

What is the minimum distance from the mirror that the fence must be positioned so that it does not block the reflection of the top of the tower from the engineer's eyes?

Two cities, A and B, are separated by a river with parallel banks. To calculate the distance between a point $P$ on the bank of city A and a point $Q$ on the bank of city B (located directly opposite), a surveyor travels $60\text{ m}$ along the bank of city A to a point $R$. From $R$, he walks another $20\text{ m}$ in the same direction to a point $S$. At $S$, he turns $90^\circ$ and moves away from the river until points $Q$ and $R$ are aligned with his current position ($T$). If the distance $ST$ is $15\text{ m}$:

1. a) Calculate the width of the river ($PQ$).
2. b) Calculate the straight-line distance between the surveyor at point $T$ and point $Q$ on the other side of the river.

A drone projects a conical beam of light onto flat ground. At a height of $10\text{ m}$, the beam illuminates a circle with a radius of $4\text{ m}$. The drone descends to a height where the illuminated area is reduced to exactly one quarter of the original area.

Draw the diagram in your notebook.

1. a) At what height is the drone located now?
2. b) If an object shaped like a cylinder with a height of $1.5\text{ m}$ and a radius of $0.85\text{ m}$ is at the center of the illuminated area, what will be the radius of the shadow cast by this object when the drone was at a height of $10\text{ m}$?

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

On a map of a forested area, a fire has affected a trapezium-shaped surface. The bases of the trapezium on the map measure $4\text{ cm}$ and $7\text{ cm}$, and its height is $3\text{ cm}$. It is known that the actual area affected is $16.5\text{ hectares}$ ($1\text{ ha} = 10\,000\text{ m}^2$).

1. a) Calculate the scale of the map.
2. b) If it is decided to reforest an adjacent area which on the map is a triangle similar to the one formed by extending the non-parallel sides of the trapezium, and whose base is the smaller base of the trapezium, what will be the actual area of this reforestation zone?

A slide is projected onto a screen located at a distance of $6\text{ m}$. The size of the projected image is $1.8\text{ m}$ high by $2.4\text{ m}$ wide.

1. a) If the original slide has dimensions of $24\text{ mm}$ by $32\text{ mm}$, at what distance from the slide is the projector's light source located?
2. b) If we move the screen so that it is $9\text{ m}$ from the projector, by how many times will the area of the image increase compared to the previous projection?

Consider an isosceles triangle $ABC$ where the unequal side $BC$ measures $12\text{ cm}$ and the equal sides measure $10\text{ cm}$ each. Inscribed in it is a square whose base lies on side $BC$ of the triangle and its other two vertices touch sides $AB$ and $AC$.

Use the similarity between the original triangle and the small triangle that lies above the square to calculate the side length of the square.

To ensure accessibility in a railway station, a continuous straight ramp is installed that bridges a vertical rise of $1.2\text{ m}$ over a horizontal distance on the ground of $9\text{ m}$. For structural reasons, a vertical support pillar must be placed at a horizontal distance of $3\text{ m}$ from the start of the ramp. It is necessary to calculate the height $h$ of said pillar and the length $d$ of the inclined section of the ramp running from the ground to the top of the pillar.

To solve it, two technicians discuss which path is more convenient:

• Strategy 1 (Direct geometric similarity): Consider the full large triangle and the small triangle under the pillar as similar in Thales' position; calculate the large hypotenuse using the Pythagorean theorem and then scale it down using the horizontal scale factor.

• Strategy 2 (Slope and local Pythagorean theorem): Calculate the constant slope of the ramp as the vertical/horizontal ratio ($1.2 / 9$), apply this slope to the $3\text{ m}$ section to directly obtain $h$, and then apply the Pythagorean Theorem only to the smaller triangle.

1. a) Solve the problem following the steps of Strategy 1 in detail.
2. b) Solve the problem following the steps of Strategy 2 in detail.
3. c) Compare both procedures: analyze which of the two avoids carrying recurring decimals or intermediate approximations, and provide a reasoned justification for which one you would recommend to minimize errors on construction plans.

To assess drought losses on an agricultural estate in Lleida, an expert uses a camera with a $36\text{ mm} \times 24\text{ mm}$ rectangular sensor and a lens with a fixed focal length of $50\text{ mm}$. The camera is mounted on a drone that points perpendicularly towards the ground, so that the lens acts as the common vertex of two inverted, similar pyramids of vision (the one going from the lens to the internal sensor and the one going from the lens to the ground outside).

1. a) If the drone flies at a fixed height of $150\text{ m}$ above the estate's ground, show why the ground covered by the photo is similar to the camera sensor and determine the linear scale factor $k$ between the real ground and the image captured on the sensor (pay attention to unit consistency).
2. b) Calculate the real dimensions in metres (length and width) of the crop rectangle that appears in a photograph and calculate the ground area covered in each shot (in square metres and in hectares).
3. c) Inverse problem: Suppose that, for a more detailed inspection, regulations require each photograph to cover a ground area of exactly $2\,160\text{ m}^2$. Using the mathematical relationship between the area ratio and the length ratio, deduce the exact height above the ground at which the drone must fly to meet this requirement.