Applications of similarity

The scale model of a new sports centre is built to a scale of $1:200$. If the Olympic swimming pool is $25 \text{ cm}$ long on the model, how many metres long will the pool be in reality?

A road map has a scale of $1:50,000$. If the distance between two neighbouring towns on the map is $12 \text{ cm}$, how many kilometres apart are they in reality?

On a floor plan of a house with a scale of $1:75$, the living room is rectangular with dimensions of $8$ cm long and $6$ cm wide. Calculate the actual surface area of the living room in square metres ($m^2$).

We have two similar rectangles. The first is $8 \text{ cm}$ wide and $12 \text{ cm}$ high. If the second has a width of $20 \text{ cm}$, what must its height be for it to maintain the same shape?

Cian wants to print a giant banner for his town's summer festival based on an original advertising poster. The original poster measures $40\text{ cm}$ wide by $60\text{ cm}$ long, and its printing cost was $8$ €. For the large banner, Cian decides to triple both dimensions (similarity ratio $k = 3$), so that it measures $120\text{ cm} \times 180\text{ cm}$. In his budget, he writes: "Since the new dimensions are triple the original ones, the amount of ink and the cost of the banner will be triple, that is, $8 \times 3 = 24$ €".\n\na) Identify the error in Cian's reasoning and explain in your own words why the cost does not triple.\nb) Using the relationship between the areas of similar figures, calculate the real cost of printing the giant banner.

An artist has made a bronze sculpture that weighs $12$ kg and is $40$ cm high. If they want to make an exact replica using the same material but with a height of $1$ metre, how much will the new sculpture weigh? (Take into account that weight is proportional to volume).

To measure the height of a tree in the school playground, we take advantage of a sunny day. A wooden batten placed vertically in the ground that is $1.5 \text{ m}$ high casts a shadow of $0.9 \text{ m}$. If at that same moment the shadow of the tree measures $6 \text{ m}$, what is the height of the tree?

A content creator is recording a video outdoors. To calculate the height of a modern sculpture, they place their tripod, which is $1.20$ metres high, so that it casts a shadow of $90$ cm on the ground. At that same moment, the shadow of the sculpture measures $4.5$ metres. How tall is the sculpture?

Calculate the height of the building ($x$) using the similarity of the triangles formed in the figure. The person's eyes are $1.60$ m from the ground, the distance from the person to the fence is $2$ m, and the distance from the fence to the building is $10$ m. The height of the fence is $2.40$ m.

In the following figure, lines $r$ and $s$ are intersected by three lines parallel to each other. Knowing that the segments on line $r$ measure $AB = 3 \text{ cm}$ and $BC = 4 \text{ cm}$, and that the corresponding segment on line $s$ is $A'B' = 4.5 \text{ cm}$, calculate the length of segment $B'C'$.

Calculate the value of the unknown segment $x$ in the following figure, knowing that the two triangles are in Thales' position:

To renovate the design of a bookcase, a carpenter places three parallel shelves intersected by two side batten rails, as shown in the figure. If the measurements of the segments on one of the battens are $AB = 25$ cm and $BC = 40$ cm, and on the other batten the segment corresponding to $AB$ measures $A'B' = 30$ cm, how long will segment $B'C'$ be?

During the modernization works at Dublin Port, a loading ramp for heavy vehicles is being installed. The side profile of the ramp has the shape of a right-angled triangle, resting on its hypotenuse on the horizontal ground. Right at the vertex of the right angle, a vertical support pillar of height $h$ is placed perpendicular to the ground, dividing the total length of the base into two segments: one of $4.5\text{ m}$ and another of $8\text{ m}$.\n\na) Model the situation by identifying which elements of the right-angled triangle represent the two segments on the ground and the height of the pillar.\nb) Calculate the exact height that the vertical pillar must have by applying the geometric mean theorem ($h^2 = m \times n$).\nc) Determine the length of the two inclined sections of the ramp (the legs) by applying the leg theorem ($b^2 = m \times c$).\nd) Check that the three sides of the overall right-angled triangle satisfy the Pythagorean theorem.

In the right-angled triangle in the figure, the hypotenuse measures $25 \text{ cm}$. The height relative to the hypotenuse, $h$, divides it into two segments, $m$ and $n$. If we know that leg $b$ measures $15 \text{ cm}$:

1. a) Calculate the length of the projection $m$ using the Cathetus Theorem.
2. b) Calculate the value of the height $h$ using the Altitude Theorem.
3. c) Find the area and perimeter of the total triangle.