Exercises: Concepts

Sara has two photographs of her dog. The first one is $10 \text{ cm}$ wide and $15 \text{ cm}$ tall. The second one is an enlargement that is $20 \text{ cm}$ wide and $30 \text{ cm}$ tall. Are these two photographs similar figures? Justify your answer by checking whether the dimensions are proportional.

Calculate the scale factor $k$ that transforms rectangle $A$ into rectangle $B$, knowing that the shorter side of $A$ measures $4 \text{ cm}$ and the shorter side of $B$ measures $12 \text{ cm}$. Is it an enlargement or a reduction?

Lucas has drawn a triangle with angles of $40^\circ$, $60^\circ$ and $80^\circ$. If he decides to create a new similar triangle by doubling the length of all its sides, what will the angles of the new triangle measure? Explain what happens to the angles of two shapes when they are similar.

On his phone screen, Marcos sees a map to a certain scale. If the scale factor between the map on the screen and reality is $k = \frac{1}{5000}$, state whether the following statements are true or false:

• a) The map is a reduction of reality.

• b) $1 \text{ cm}$ on the map represents $5000 \text{ cm}$ in reality.

• c) The angles between streets change on the map so that they fit on the screen.

A logo is in the shape of a regular pentagon with a side of $3 \text{ cm}$. For a shop front, a similar logo is made with a scale factor of $k = 15$. How long will the side of the new logo be?

Look at the following pairs of rectangles. Without doing complex calculations, state which one cannot be similar to the original and explain why by looking at its shape.

If we have two similar polygons with a scale factor of $k = 2,5$, and the perimeter of the small polygon is $20 \text{ cm}$, what will the perimeter of the larger polygon be? Remember the relationship between the scale factor and the perimeters.

Draw a letter "L" on a grid that is $2$ squares wide and $4$ squares high. Next, draw another letter "L" beside it that is similar but with a scale factor of $k = 0,5$. How many squares high will the new shape be?

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

Carla has a $20\text{ cm} \times 30\text{ cm}$ rectangular art sheet for her Visual Arts class and uses exactly one $60\text{ ml}$ tube of poster paint to cover it completely. For an exhibition at school, she wants to make an enlarged, similar version by multiplying the length of all its sides by $3$ (that is, with a scale factor of $k = 3$). Her classmate Hugo tells her: "Since the sides are three times as long, you will need three times as much paint, so buy $3$ tubes of $60\text{ ml}$ ($180\text{ ml}$) and you'll have enough."\na) Point out the error in Hugo's reasoning and explain, using the properties of similar figures, why the amount of paint is not proportional to the scale factor $k$.\nb) Calculate how many millilitres of poster paint and how many whole tubes of paint Carla will actually need to paint the large sheet.

The city council is going to paint a children's multi-sports court on the ground of a park that will be similar to a regulation futsal court. The regulation court is $40\text{ m}$ long by $20\text{ m}$ wide. The children's court will be $20\text{ m}$ long.\na) Without doing detailed calculations yet, make a reasoned estimate: will the perimeter of the children's court be half or a quarter of the regulation court? And its surface area, will it be half, a quarter, or an eighth?\nb) Now calculate the scale factor $k$, the perimeter of both courts, and the area of both courts.\nc) Check whether your estimates from part a) match the calculated values and justify the mathematical relationship obtained between the scale factor, the perimeters, and the areas.

An illustrator is designing two versions of the same rectangular logo. The original logo has dimensions of $12\text{ cm}$ in length and $8\text{ cm}$ in width. If they want to create an enlarged version that is similar and whose length measures $30\text{ cm}$:

1. Calculate the scale factor $k$ of the enlargement.

2. Determine the width of the new logo.

3. If the perimetre of the small logo is $40\text{ cm}$, what will the perimetre of the large one be? Use the relationship between the perimetres of similar figures to check your answer.

Calculate the value of the unknown line segment $x$ in the following figure, given that lines $r$, $s$ and $t$ are parallel. Apply Thales' Theorem and express the result to two decimal places.

Two triangles, $ABC$ and $A'B'C'$, are in Thales' position. The small triangle has sides measuring $AB = 5\text{ cm}$, $BC = 6\text{ cm}$ and $AC = 4\text{ cm}$. If the corresponding side $AB'$ of the larger triangle measures $12,5\text{ cm}$:

1. What is the scale factor between the larger triangle and the smaller one?

2. Find the length of sides $B'C'$ and $AC'$.

3. Check that the quotient of their areas is equal to the square of the scale factor.

An architect has drawn the floor plan of a room to a scale of 1:50. On the plan, the room is a rectangle $12\text{ cm}$ long by $8\text{ cm}$ wide.

1. Calculate the actual dimensions of the room in metres.

2. The architect wants to place a rug that occupies $6\text{ m}^2$ in reality. What area will the rug occupy on the plan (in $\text{cm}^2$)?

3. If they decided to change the scale to 1:100, how would the measurements of the drawing change compared to the previous plan?

To measure the height of an electricity pylon, a technician observes that at a particular moment the pylon casts a shadow of $18\text{ metres}$. At the same time, they place a vertical stick $1,5\text{ metres}$ high, which casts a shadow of $2,25\text{ metres}$.

1. Justify why the triangles formed by the pylon and its shadow, and the stick and its shadow, are similar.

2. Calculate the height of the pylon.

Determine whether the following pairs of triangles are similar and state which similarity criterion you used in each case:

• Case A: Triangle 1 has angles of $40^\circ$ and $75^\circ$. Triangle 2 has angles of $75^\circ$ and $65^\circ$.

• Case B: Triangle 1 has sides of $6\text{ cm}$, $9\text{ cm}$ and $12\text{ cm}$. Triangle 2 has sides of $8\text{ cm}$, $12\text{ cm}$ and $16\text{ cm}$.

• Case C: Both triangles have an angle of $50^\circ$. In the first, the sides forming this angle measure $4\text{ cm}$ and $7\text{ cm}$. In the second, they measure $8\text{ cm}$ and $14\text{ cm}$.

In a right-angled triangle, the altitude to the hypotenuse divides it into two segments (projections of the legs) measuring $3,6\text{ cm}$ and $10\text{ cm}$.

1. Calculate the altitude of the triangle using the Geometric Mean Theorem.

2. Calculate the length of the legs using the Cathetus Theorem.

3. Draw a diagram of the triangle showing all the values found.

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

We have two cylindrical juice cartons that are similar. The small carton has a height of $10\text{ cm}$ and a capacity of $250\text{ ml}$. The large carton has a height of $20\text{ cm}$.

1. What is the scale factor $k$ between the large carton and the small one?

2. Calculate the capacity (volume) of the large carton in litres. Remember the relationship between the scale factor and the volume of similar solids.

3. If the base area of the small one is $25\text{ cm}^2$, what will the base area of the large one be?

The interior courtyard of a cultural centre in your autonomous community is shaped like a trapezium with a total area of $54\text{ m}^2$ and its longest side measures $12\text{ m}$. The staff at the front desk want to order a similar scale model of the courtyard to display it in a glass showcase. Due to the dimensions of the showcase, the area of the model must be exactly $6\text{ m}^2$.\na) Knowing that the relationship between the areas of similar figures depends on $k^2$, write the corresponding equation and calculate the scale factor $k$ of the model relative to the real courtyard.\nb) Determine how long the side corresponding to the longest side of $12\text{ m}$ should be in the model.\nc) If the full perimeter of the real courtyard is $36\text{ m}$, how many centimetres of wooden moulding will be needed to edge the perimeter of the model?

In a graphic design workshop, they are preparing a triangular banner for a campaign promoting the use of public transport. The initial sketch is a right-angled triangle whose legs measure $6\text{ cm}$ and $8\text{ cm}$. For the final mount, they need a similar banner with an enlargement scale factor of $k = 4$. To find the surface area of fabric that the large banner will occupy, two designers suggest different methods:\n- Lucía's method: First calculate the new lengths of the legs by multiplying them by $4$ and, using them, calculate the area of the large triangle using the formula $(\text{base} \cdot \text{height}) / 2$.\n- Mateo's method: First calculate the area of the initial sketch and multiply it directly by the scale factor squared ($k^2$).\na) Solve the problem applying both methods step by step and verify numerically whether the same result is reached.\nb) If the figure were not a simple triangle, but a logo in the shape of an irregular $7$-sided polygon whose initial area is already known ($35\text{ cm}^2$), which of the two methods would you choose and why? Justify your choice by assessing the amount of data and calculations required.

A perfume company launches two sizes of its flagship bottle, which have the shape of similar square-based pyramids. The "Large" bottle has a base area of $144 \text{ cm}^2$ and a height of $15 \text{ cm}$. The "Mini" bottle has been designed so that its total surface area of glass is exactly one ninth of the surface area of the "Large" bottle. If the cost of the perfume is $0,25$ € per $\text{cm}^3$, calculate the price difference between the contents of both bottles.

A street artist wants to project the shadow of a vertical sculpture $2,4 \text{ m}$ high onto a wall that is $6 \text{ m}$ away. To do so, they place a spotlight on the ground at a distance $d$ from the sculpture. If they want the projected shadow on the wall to be exactly $8 \text{ m}$ high, at what distance from the sculpture should they place the spotlight? Use Thales' Theorem to justify your answer and draw a diagram of the similar triangles.

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

In a right-angled triangle $ABC$ (right-angled at $A$), the altitude $h$ to the hypotenuse is drawn, dividing it into two segments $m$ and $n$. It is known that leg $b$ measures $15 \text{ cm}$ and its projection on the hypotenuse, $n$, measures $9 \text{ cm}$. Calculate the perimetre and area of the original triangle using the Leg and Altitude Theorems.

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

A house floor plan is drawn to a scale of $1:75$. On this plan, the kitchen is rectangular, measuring $4 \text{ cm} \times 6 \text{ cm}$. The owner wants to renovate the floor with square tiles measuring $40 \text{ cm}$ on each side.

1. a) Calculate the real area of the kitchen in square metres.
2. b) How many whole tiles are needed, at least, to cover the real kitchen floor? (Note that offcuts can be reused, but tiles can only be bought whole).

Two regular pentagons are similar. The perimetre of the first one is $75 \text{ cm}$. It is known that the area of the second one is $1,44$ times the area of the first one.

1. a) Determine the scale factor $k$ between the second pentagon and the first one.
2. b) Calculate the perimetre of the second pentagon.
3. c) If the apothem of the first pentagon measures $10,32 \text{ cm}$, what will the radius of the circumscribed circle of the second pentagon measure? (Round to two decimal places).

A $50$-inch television has an aspect ratio of $16:9$ (this means that the base and height are proportional to $16$ and $9$). Knowing that $1$ inch is equivalent to $2,54 \text{ cm}$ and that the "inches" of a television refer to the measurement of its diagonal:

1. a) Calculate the real dimensions (width and height) of the television in centimetres.
2. b) If an image of $800 \times 450$ pixels is shown full screen without distortion, what is the area of each pixel on the actual screen?

A scale model of a skyscraper made to a scale of $1:250$ has a volume of $320 \text{ cm}^3$ and its base has an area of $20 \text{ cm}^2$.

1. a) Calculate the real height of the skyscraper in metres.
2. b) If the real building is to be covered with glass panels costing $120$ € per square metre, calculate the total cost of covering the facade (lateral surface area), knowing that the skyscraper is in the shape of a right prism and that the scale model has a lateral surface area of $180 \text{ cm}^2$.

A municipality in southern Spain is implementing a water-saving plan due to a drought and needs to model the water consumption of the pools in its municipal sports centre. They have two right prism-shaped pools that are similar to each other (they have the same shape and proportions in length, width, and depth): the paddling pool and the adult recreational pool. The paddling pool is $10\text{ m}$ long and has a capacity of $30\text{ m}^3$ of water ($30.000\text{ litres}$). The adult pool is $25\text{ m}$ long.\na) Model the situation by identifying the scale factor $k$ that transforms the dimensions of the paddling pool into those of the adult pool.\nb) Calculate the total volume of water, in cubic metres and in litres, needed to fill the adult pool.\nc) Before filling them, a protective resin coating must be applied to the bottom and the four interior walls. If $16\text{ kg}$ of resin were used for the paddling pool, calculate how many kilograms will be needed for the adult pool.\nd) Explain why the increase in paint/resin consumption does not match the increase in water consumption when comparing both facilities.

A master chocolatier from Astorga makes solid, similar figures shaped like traditional bells. The small figure is $10\text{ cm}$ tall, weighs $125\text{ g}$, and sells for €$3,50$. For a special order from a religious brotherhood, he wants to make a giant similar bell that weighs exactly $1\text{ kg}$ ($1.000\text{ g}$). Both figures are made from the same type of dark chocolate, so their mass is directly proportional to their volume.\na) Using the relationship between the masses (volumes), deduce the value of the linear scale factor $k$ between the giant bell and the small one.\nb) How tall will the giant bell be?\nc) Metallic cellophane paper is used to wrap it. If the small bell requires $180\text{ cm}^2$ of paper to be completely covered, calculate the surface area of paper needed to wrap the giant bell.\nd) The customer claims: "Since the height of the giant bell is twice that of the small one, the fair price should be double: €$7,00$". Justify mathematically whether the customer's claim is correct or not, and explain what the price of the giant bell should be if it is charged proportionally to the amount of chocolate used.