Exercises: Exercises and 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?

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 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?

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?

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 decorate a wall, we want to enlarge a photograph measuring $10 \text{ cm} \times 15 \text{ cm}$ until its longer side measures $1.2 \text{ metres}$. How long will the shorter side of the enlarged photograph be?

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?

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

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.

You are hiking through Killarney National Park (Co. Kerry) and you want to find out the height of a wooden observation tower without climbing it. At $11:30$ in the morning, you stick a $1.20\text{ m}$ tall hiking pole into the horizontal ground, which casts a shadow of $80\text{ cm}$. At that very same moment, you measure the shadow cast by the tower on the flat ground and get $6.4\text{ m}$.\n\na) Before performing any formal numerical calculation, make a reasoned estimate: will the height of the tower be greater or less than the length of its shadow? Do you think it will exceed $10\text{ m}$?\nb) Set up the similarity proportion between the heights and the shadows to calculate the exact height of the observation tower.\nc) Compare the value obtained with your initial estimate and evaluate whether the result is consistent with reality.

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?

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 ($m^2$).

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 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?

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).

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 kilometres?

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.

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 $1.2$ metres:

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?

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.

A landscape architecture studio in Galway needs to design two strictly similar triangular lawn areas. The small flowerbed has an area of $12\text{ m}^2$ and its longest side measures $6\text{ m}$. The city council's technical specifications mandate that the large flowerbed must have an area of exactly $108\text{ m}^2$.\n\na) Determine the ratio between the areas of both flowerbeds and deduce from it what the linear similarity ratio $k$ between their sides must be.\nb) How long will the longest side of the large flowerbed be?\nc) If $16\text{ m}$ of metal edging were used to surround the entire perimeter of the small flowerbed, calculate how many meters of edging will be needed to fence the large flowerbed. Justify your answer without needing to calculate the measure of the other two individual sides.

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?

To measure the height of the Spire in Dublin 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 centre 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?

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

On a map with a scale of $1:25,000$, a nature reserve is in the shape of a trapezium. 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 kilometres ($\text{km}^2$).
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 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?

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.

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)?

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 centre 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 Pythagoras' 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?

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 design of a wooden right-triangular truss for the roof of a bioclimatic home, the vertical reinforcing beam coincides with the altitude to the hypotenuse. The lower horizontal beam (the hypotenuse) measures $25\text{ m}$ and the projection of the shorter rafter onto the ground measures $9\text{ m}$. The exact length of the vertical reinforcing beam ($h$) needs to be calculated.\n\nTo solve it, two architecture students propose different approaches:\n- Aoife says: "First I calculate the other projection on the ground by subtracting from the total hypotenuse ($n = 25 - 9 = 16\text{ m}$) and then I directly calculate $h$ with the geometric mean theorem".\n- Niamh says: "First I calculate the shorter rafter (the leg) with the leg theorem, and once I have that side, I apply the Pythagorean theorem to the right triangle formed by that leg, its projection of $9\text{ m}$, and the altitude $h$".\n\na) Fully develop both methods with their corresponding formulas and verify whether they arrive at the same value for $h$.\nb) Compare the two procedures and critically justify which of the two is preferable in terms of efficiency of operations and a lower probability of making rounding or calculation errors.\nc) If on another roof we did not know the total hypotenuse, but only the length of the leg and its projection, could Aoife's method still be used without first calculating the hypotenuse? Justify your answer.

Ordnance Survey Ireland (OSi) publishes maps at a scale of $1:25\,000$. A fire prevention team from the Department of Agriculture is analyzing a forested area in the Wicklow Mountains shaped like a right-angled triangle on the map. On the official map, the hypotenuse of this triangle measures $10\text{ cm}$ and the projection of one of its legs onto the hypotenuse measures $3.6\text{ cm}$.\n\na) Using the geometric mean theorem and the leg theorem directly on the measurements from the map, find the lengths of the two legs and the altitude to the hypotenuse (in paper centimeters).\nb) Calculate the area of the forest zone on the map (in $\text{cm}^2$). Next, determine the actual surface area of this zone in hectares ($1\text{ ha} = 10\,000\text{ m}^2$) exclusively using the similarity ratio between areas $k^2$, without first converting each side to real meters. Explain why this procedure is more direct.\nc) If the forest brigades' clearing protocol requires creating a safety perimeter strip and they advance at an average rate of $1.2$ linear $\text{km}$ per day, how many full days of work will they need to cover and secure the entire real perimeter of the triangular site?