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?

A smartphone with a $6 \text{ pulgadas}$ diagonal has a screen measuring $13.2 \text{ cm}$ high and $7.4 \text{ cm}$ wide. If we want to view the same content on a tablet that is exactly twice as large (scale factor $k=2$), what will be the dimensions of the tablet's screen?

Two television screens are similar. The first has a diagonal of $32$ inches and its surface area is approximately $2800$ $cm^2$. If the second screen has a diagonal of $48$ inches:

1. What is the scale factor between the large screen and the small one?

2. What surface area does the $48$ inch screen occupy?

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?

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 $B'C'$.

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.