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.

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.

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.

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 $70^\circ$.

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 ($1\text{ km}^2 = 100\text{ ha}$).

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.

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.

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.

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 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?

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?

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?

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.

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).

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?

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?

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 $1:n$).