Exercises and Applications of Similarity

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

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

Two towns, A and B, are $18$ km apart in reality. On a road map, this distance is represented by a segment of $12$ cm.

1. Calculate the scale of the map.

2. If two other towns are separated on the same map by $7.5$ cm, what is their actual distance in kilometers?

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?

A rectangular photograph of $10 \times 15$ cm is to be enlarged to make a poster. If we want the longer side of the enlargement to be $1.2$ meters:

1. How long will the short side of the poster be?

2. How many times larger is the area of the poster than the area of the original photograph?

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 meters ($m^2$).

On a map with a scale of $1:25,000$, a nature reserve is in the shape of a trapezoid. On the paper, its bases measure $4 \text{ cm}$ and $6 \text{ cm}$, and its height is $3 \text{ cm}$.

1. a) Calculate the area of the reserve on the map in $\text{cm}^2$.
2. b) Using the scale factor of the areas, calculate the actual area of the reserve in square kilometers ($\text{km}^2$).
3. c) If the government decides to expand the reserve by increasing all its linear dimensions by $20\%$, what will be the new actual surface area?

A video game designer is creating two models of a defensive tower. The small model has a lateral surface area of $150 \text{ cm}^2$ and a volume of $250 \text{ cm}^3$. If the large model must have a lateral surface area of $600 \text{ cm}^2$ so that the details look clear:

1. a) What is the scale factor $k$ between the lengths of both models?
2. b) What will be the volume of the large model?
3. c) If 40 grams of resin were needed to make the small model, how many kilograms of resin will be needed to make 10 large models?

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 $1$ meter, 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 $1.20$ meters high, so that it casts a shadow of $90$ cm on the ground. At that same moment, the shadow of the sculpture measures $4.5$ meters. 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.

To measure the height of the Torre Latinoamericana in Mexico City without using complex instruments, a group of students uses a mirror placed on the ground. The tallest student, who is $1.85 \text{ m}$ tall, stands at a distance such that he sees the top of the tower reflected in the center of the mirror. The mirror is $120 \text{ m}$ from the base of the tower and the student is $2.3 \text{ m}$ from the mirror. If the student's eyes are $10 \text{ cm}$ below the top of his head:

1. a) Draw a diagram of the situation identifying the similar triangles.
2. b) Calculate the height of the tower knowing that the angle of incidence of light is equal to the angle of reflection.
3. c) If another student of height $1.60 \text{ m}$, whose eyes are also $10 \text{ cm}$ below the top of their head, wanted to see the same thing without moving the mirror, at what distance from the mirror should they stand?

A pinhole camera is a closed box with a small hole through which light enters, projecting the image of an outside object onto the opposite wall. If we have a box $25 \text{ cm}$ deep and we want to photograph a tree $12 \text{ m}$ high that is at a distance of $30 \text{ m}$ from the camera:

1. a) What will be the height of the tree's image projected inside the box?
2. b) If we move the camera closer to the tree until the projected image is twice as large, how far from the tree will we be?
3. c) What is the scale factor between the real object and the image in this last case?

Two vertical buildings are separated by a street $20 \text{ m}$ wide. The taller building measures $45 \text{ m}$ and the shorter one $30 \text{ m}$. From the roof of the shorter building, the base of the tall building is observed. In that same line of sight, a lamppost stands $5 \text{ m}$ from the shorter building.

1. a) How tall must the lamppost be for its top to be exactly on that line of sight?
2. b) If we run a straight cable from the roof of the tall building to the roof of the short building, how long will the cable be?
3. c) At what height above the ground would that cable cross the lamppost (assuming the lamppost is tall enough)?

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?

A triangular plot of land with a surface area of $300 \text{ m}^2$ is to be divided into two plots using a fence parallel to one of its sides. The fence is placed so that it cuts the other two sides at one third of the distance from the top common vertex.

1. a) What is the scale factor between the small (upper) triangle and the original plot of land?
2. b) Calculate the area of each of the two resulting plots.
3. c) If the base of the original plot of land measured $30 \text{ m}$, how long will the new fence be?

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.

A park is in the shape of a regular hexagon with a side length of $40 \text{ m}$. A circular fountain is to be built in its center that is similar to a small fountain $2 \text{ m}$ in diameter located at the entrance. If the scale factor between the radius of the circle inscribed in the hexagon (apothem) and the radius of the fountain must be $4$:

1. a) Calculate the apothem of the hexagon (you can use the Pythagorean theorem).
2. b) Calculate the diameter of the new central fountain.
3. c) What is the ratio of the area of the park to the area of the central fountain?