Exercises: Similarity of figures

Look at the two similar polygons in the figure. Calculate the lengths of the three unknown sides of the second polygon, knowing that the side corresponding to $3 \text{ cm}$ measures $6 \text{ cm}$.

Determine whether two rectangles are similar, knowing that the dimensions of the first are $15 \text{ cm}$ base by $10 \text{ cm}$ height, and those of the second are $24 \text{ cm}$ base by $16 \text{ cm}$ height. Justify your answer.

On a map drawn to a scale of $1:25 000$, the straight-line distance between two villages is $8 \text{ cm}$. Calculate the actual distance between both villages in kilometres.

The actual straight-line distance between two cities is $120 \text{ km}$. If we want to represent them on a map to a scale of $1:500 000$, how far apart will they be on the map, in centimetres?

Two triangles are similar with a scale factor of $k = 3$. If the perimeter of the smaller triangle is $14 \text{ cm}$, what is the perimeter of the larger triangle?

The area of a square is $16 \text{ cm}^2$. If we construct a similar square using a scale factor of $k = 2,5$, what will the area of the new square be?

A packing box in the shape of a cuboid has a volume of $50 \text{ dm}^3$. If we make a similar box whose dimensions are exactly double those of the original, what will its volume be?

The areas of two regular pentagons are $25 \text{ cm}^2$ and $100 \text{ cm}^2$. What is the scale factor of the sides of the larger pentagon relative to the sides of the smaller pentagon?

Marcos wants to order an advertising banner for his town's local festival, based on an original paper poster. The original design is $40\text{ cm}$ wide by $60\text{ cm}$ high, and the print shop charged him $6\text{ \euro}$. To display it on the town square stage, Marcos decides to order a similar banner, tripling all linear dimensions (similarity ratio $k = 3$). When calculating the necessary budget, Marcos states: "Since the sides are multiplied by $3$, the price of the banner will be triple that of the small poster, that is, $3 \cdot 6 = 18\text{ \euro}$".

1. a) Identify the conceptual mistake made by Marcos and explain in detail why it is not correct to apply the linear scale ratio $k$ to the cost of the material.
2. b) What scale ratio must be applied to determine the change in the surface area of the banner relative to the small poster?
3. c) Knowing that the printing cost is proportional to the surface area of the material used, calculate the actual price that Marcos should budget for the giant banner.

According to the official regulations of the Spanish Basketball Federation (FEB), a basketball court has dimensions of $28\text{ m}$ in length by $15\text{ m}$ in width. A town council is going to display a similar flat scale model of the facilities at a $1:50$ scale.

1. a) Before carrying out exact numerical calculations, mentally estimate whether the surface area of the scale model will be larger, smaller, or similar to the area of a standard DIN A4 sheet of paper (whose approximate dimensions are $21\text{ cm} \times 30\text{ cm}$, that is, about $630\text{ cm}^2$). Justify your prediction.
2. b) Express the dimensions of the actual court in centimetres and calculate the length and width of the scale model.
3. c) Find the area of the scale model in square centimetres and check whether your initial estimation was correct (indicate approximately how many DIN A4 sheets its surface area is equivalent to).
4. d) Verify that the ratio between the area of the actual court and the area of the scale model is exactly equal to $k^2$.

In a pizzeria, a small pizza with a diameter of $20 \text{ cm}$ costs $6 \text{ \euro}$. If the price is strictly proportional to the area of the pizza, how much should a family pizza with a diameter of $30 \text{ cm}$ cost?

A scale model of a skyscraper is built to a scale of $1:200$. If the volume of the model is $1,5 \text{ dm}^3$, calculate the actual volume of the skyscraper in cubic metres.

The sides of a quadrilateral measure $4$, $6$, $7$ and $9 \text{ cm}$. A quadrilateral similar to this has a total perimeter of $65 \text{ cm}$. Calculate the length of each side of the larger quadrilateral.

A cylindrical paint tin has a capacity of $2 \text{ litres}$ and its height is $15 \text{ cm}$. A similar tin is to be made with a capacity of $16 \text{ litres}$. What will the height of the new tin be?

On a topographic map to a scale of $1:10 000$, an agricultural plot occupies an area of $12 \text{ cm}^2$. Calculate the actual area of the plot in hectares. (Remember that $1 \text{ hectárea} = 10 000 \text{ m}^2$).

Look at the attached plan. Knowing that the shaded plot has an area on paper of $12 \text{ cm}^2$ and that the linear scale indicates $1:5000$, calculate the actual area of the plot in square metres.

To paint the front of a doll's house, $50 \text{ ml}$ of paint was used. If we want to paint the front of a real house that is $12$ times larger in all its linear dimensions, how many litres of paint will we need, assuming that both fronts are similar figures?

Two spheres are similar. The surface area of the larger sphere is $9$ times the surface area of the smaller one. If the volume of the smaller sphere is $4 \text{ cm}^3$, what is the volume of the larger sphere?

A sustainable packaging company manufactures three models of gift boxes with similar polyhedral shapes: Model A (small), Model B (medium), and Model C (large). The design team only has relative surface area and volume data between them:

• To wrap the outside of box B, exactly $2,25$ times more paper is needed than to wrap box A (the ratio between their total surface areas is $2,25$).

• The internal capacity of box C is exactly $3,375$ times the volume of box A.

1. a) Deduce and calculate the linear similarity ratio $k$ of box B with respect to box A, and the linear similarity ratio $k$ of box C with respect to box A. Based on the results, what geometric relationship exists between box B and box C?
2. b) If the main edge of box A measures $12\text{ cm}$, calculate the lengths of the corresponding edges of boxes B and C.
3. c) If box A requires $240\text{ cm}^2$ of cardboard and holds a volume of $400\text{ cm}^3$, calculate directly (using the corresponding area and volume ratios) the cardboard needed to manufacture box C and the capacity in $\text{cm}^3$ of box B.

At an olive cooperative in Jaén, extra virgin olive oil (EVOO) is packaged in geometrically similar cylindrical metal cans. They have two formats: a $1\text{-litre}$ individual can and a $5\text{-litre}$ family can. For the $1\text{-litre}$ container, the height is $18\text{ cm}$ and $587\text{ cm}^2$ of tinplate is needed to manufacture its total surface area.

1. a) Knowing that capacity measures volume ($1\text{ litre} = 1\text{ dm}^3$), calculate the linear similarity ratio $k$ between the family can and the individual can (calculate $k = \sqrt[3]{5}$ rounding to two decimal places).
2. b) Calculate the height of the $5\text{-litre}$ family can. Is the height multiplied by $5$ when quintupling the capacity? Justify your answer based on the properties of three-dimensional similarity.
3. c) A worker believes that five times as much tinplate ($5 \cdot 587 = 2935\text{ cm}^2$) will be needed to manufacture the $5\text{-litre}$ can. Calculate the actual surface area of tinplate needed by applying $k^2$ and explain why it is more cost-effective in terms of material for the cooperative to package in the large format rather than the small one.

Give reasons to show whether the following statement is true or false: "All rectangles are similar to each other, since all their interior angles measure $90^\circ$." What happens in the case of rhombuses? Justify your answer using the conditions for similarity of polygons.

When enlarging a rectangular photograph in an editing program, its area increased by $69\%$. Determine the scale factor $k$ applied in the enlargement and calculate the percentage by which the perimeter of the photograph has increased.

A titanium engineering component weighs $4,5 \text{ kg}$. A geometrically similar component of reduced size is to be manufactured, which must weigh exactly $500 \text{ grams}$. If the original component has a maximum length of $30 \text{ cm}$, what will the maximum length of the new reduced component be?

The base area of a pyramid is $45 \text{ cm}^2$ and its volume is $180 \text{ cm}^3$. The pyramid is cut by a plane parallel to the base, obtaining a frustum of a pyramid and a smaller pyramid at the top. If the base area of the smaller pyramid is $20 \text{ cm}^2$, calculate its volume.

Look at the figure where a cone of height $H$ and volume $V$ is cut by a plane parallel to the base at a height $h$ measured from the vertex. If we want the volume of the small cone to be exactly half the volume of the large cone ($\frac{V}{2}$), what must the exact value of the ratio $\frac{h}{H}$ be? Express the answer using surds.

A cylindrical water tank costs $120 \text{ \euro}$ to paint on the outside, and filling it with water costs $400 \text{ \euro}$. A similar tank is constructed whose height is $50\%$ greater. Calculate the total cost of painting and filling the new tank, assuming that the rates per square metre of paint and per litre of water remain constant.

A company manufactures similar cylindrical tins. Tin A requires $150 \text{ cm}^2$ of tinplate for its manufacture and contains $300 \text{ ml}$ of juice. If tin B requires $540 \text{ cm}^2$ of tinplate to be manufactured, how many millilitres of juice does it contain?

A sequence of 2D shapes is constructed as follows: the first shape has an area of $100 \text{ cm}^2$. Each subsequent shape is similar to the previous one with a scale factor $k = \frac{3}{4}$. Write down the expression for the area of the $n$th shape and calculate from which shape onwards the area will be strictly less than $10 \text{ cm}^2$.

A town council plans to build a new triangular landscaped area that is similar to an existing children's playground. In the current playground, which has the shape of a right-angled triangle, the legs measure $15\text{ m}$ and $20\text{ m}$, and the hypotenuse measures $25\text{ m}$. The new square must have a planned total surface area of $1500\text{ m}^2$. To order the perimeter fence, the technicians need to know the exact perimeter of the new square.

Two calculation proposals arise in the technical office:

• Method 1 (Solving for sides): Find the area of the current playground, set up a system of proportions to calculate the new base and the new height individually, obtain the new hypotenuse using the Pythagorean theorem, and add the three sides together.

• Method 2 (Global use of similarity ratios): Find the area of the current playground, calculate the area ratio $k^2$, obtain the linear ratio $k = \sqrt{k^2}$, calculate the perimeter of the current playground, and multiply it by $k$.

1. a) Fully work out the solution using Method 2 and find the metres of fencing needed.
2. b) Justify in detail why Method 2 is more efficient and introduces fewer rounding errors than Method 1.
3. c) If the plot were an irregular $7$-sided polygon with no known right angles instead of a right-angled triangle, explain why Method 1 would be unfeasible with the given data while Method 2 would apply in exactly the same way.

For an exhibition on transport at the Museo del Ferrocarril in Madrid, Renfe orders a $1:87$ scale model (the classic H0 railway scale) of an Avril (S-106) high-speed passenger car. The actual carriage is $20\text{ m}$ long, has an external sheet metal surface area of $180\text{ m}^2$, and a usable interior volume of $75\text{ m}^3$.

1. a) Calculate the length of the model carriage in centimetres, rounding to one decimal place.
2. b) To paint the exterior of the model, a jar of modelling paint is purchased that covers a maximum of $350\text{ cm}^2$. Convert the actual area of the carriage to $\text{cm}^2$, apply the surface scale, and reasonably justify whether a single jar of paint will be enough to paint the entire model.
3. c) A museum visitor states: "Since the scale is $1:87$, the volume of air that fits inside the model will be $75 / 87 \approx 0,86\text{ m}^3$, that is, about $860\text{ litres}$". Prove by calculating the volume of the model (in $\text{cm}^3$) why that statement represents physical nonsense, and determine approximately how many millilitres the actual capacity of the scale model is equivalent to.