Exercises: Similar shapes

A graphic designer is creating two rectangular posters that must be similar. The original poster measures $60 \text{ cm}$ wide and $40 \text{ cm}$ high. If the second poster has a width of $120 \text{ cm}$, what must its height be for it to keep the same shape? Also calculate the scale factor $k$ of the enlargement.

We have two photographs of the same monument, a small one and an enlarged one. We know they are similar figures. In the small photo, the height of the monument is $5 \text{ cm}$ and its width is $3 \text{ cm}$. In the enlarged photo, the height is $15 \text{ cm}$.

1. a) What will the width of the monument be in the enlarged photo?

1. b) Give the similarity ratio (from the small photo to the enlarged one) as a decimal number.

For an art project, Lucía drew a lightning bolt on a grid. Now she needs to draw another lightning bolt that is a reduction of the first with a scale factor of $k = 0.5$. If the original lightning bolt measures $8 \text{ cm}$ long, how long will the new lightning bolt measure? What does it mean for the scale factor to be less than $1$?

Two pentagons are similar and the scale factor between them is $k = 3$. If the perimeter of the smaller pentagon is $25 \text{ cm}$, what is the perimeter of the larger pentagon? Explain your answer by stating the relationship between the scale factor and the perimeters of similar shapes.

We have three right triangles, whose legs have the following measures:

• Triangle A: legs of $3\text{ cm}$ and $4\text{ cm}$.

• Triangle B: legs of $6\text{ cm}$ and $8\text{ cm}$.

• Triangle C: legs of $5\text{ cm}$ and $10\text{ cm}$.

Identify which of them are similar by checking whether their corresponding sides are proportional. Calculate the similarity ratio between the two you choose.

A map of a nature reserve is drawn to a scale such that $2 \text{ cm}$ on paper represents $500 \text{ m}$ in reality. If the main trail measures $6 \text{ cm}$ on the map, what is its actual length in meters? Also express the scale of the map in the usual format $1:n$.

A rectangular tablet screen has dimensions of $24 \text{ cm} \times 18 \text{ cm}$. We want to manufacture a case for a smaller smartphone model that is similar to the tablet. If the longer side of the smartphone measures $12 \text{ cm}$, how long should the shorter side measure? What is the reduction percentage applied?

Are two squares similar if one has a side of $5 \text{ cm}$ and the other a side of $12 \text{ cm}$? Justify your answer based on the two conditions that polygons must satisfy to be similar: corresponding angles and corresponding sides.

A graphic designer is working on the logo for a new app. The original design fits into a square with a side length of $4\text{ cm}$. For the website header, they need to enlarge it so that the new side length is $10\text{ cm}$.

1. a) Calculate the scale factor $k$ of the enlargement.
2. b) If the original logo has an internal decorative line measuring $2.5\text{ cm}$, how long will that same line measure in the enlarged logo?

In Art class, Lucía drew a lighthouse on a grid. Now she wants to reproduce it at three times its original size to decorate a mural.

1. a) If in the original drawing the lighthouse has a base of $2\text{ cm}$, what width should the base have in the reproduction?
2. b) Lucía calculated that the area occupied by the lighthouse in her notebook is $12\text{ cm}^2$. What area will the lighthouse occupy on the mural? Remember the relationship between the areas of similar shapes.

Two rectangular photo frames are similar. The small frame has dimensions of $10\text{ cm} \times 15\text{ cm}$. The large frame has a perimeter of $150\text{ cm}$.

1. a) What is the scale factor between the large frame and the small one?
2. b) What are the dimensions (length and width) of the large frame?

On a video game map of an island, the distance between "Pirate Port" and "Mount Destiny" is $4.5\text{ cm}$. If we know that the scale of the map is $1:20,000$:

1. a) What is the actual distance, expressed in meters, between these two points?
2. b) If a forest occupies an area of $3\text{ cm}^2$ on the map, what is its actual surface area in square meters?

A shopping center has a floor plan in the shape of a trapezoid. On the architectural plan, drawn to a scale of $1:500$, the longer base measures $16\text{ cm}$, the shorter base measures $12\text{ cm}$ and the height is $8\text{ cm}$.

1. a) Calculate the actual dimensions of the shopping center floor plan in meters.
2. b) Calculate the ratio between the area on the plan and the actual area.

Two "STOP" road signs have been made that are similar (regular octagons). The small sign has a side of $20\text{ cm}$ and an area of approximately $1931\text{ cm}^2$. If the large sign has a side of $30\text{ cm}$:

1. a) What is the scale factor?
2. b) What is the perimeter of the large sign?
3. c) What will the area of the large sign be? (Express the result in $\text{cm}^2$).

A film crew uses a scale model of a skyscraper for an action scene. The model measures $1.2\text{ m}$ in height and its base has a surface area of $0.15\text{ m}^2$. If the real skyscraper has a height of $180\text{ m}$:

1. a) Calculate the scale to which the model was built.
2. b) What is the actual surface area of the base of the building?

Look at the two television screens in the figure. Although they have different sizes, they are said to maintain the same aspect ratio (they are similar). The small screen is $32$ inches and the large one is $50$ inches.

1. a) If the width of the $32$-inch screen is $70.8\text{ cm}$, what is the width of the $50$-inch screen?
2. b) If the area of the small screen is $2\,827\text{ cm}^2$, what is the area of the large screen?

A graphic design student is working with a rectangular logo that measures $12\text{ cm}$ wide and $8\text{ cm}$ high. First, they decide to apply a 150% enlargement for an advertising poster. Then, based on that poster, they perform a reduction to 40% to fit it onto a business card.

• a) What is the total scale factor between the original logo and the business card logo?

• b) Is the business card logo larger or smaller than the original? Explain your answer by calculating the final dimensions.

• c) If the area of the original logo was $96\text{ cm}^2$, what will the area of the logo on the business card be?

On a model of a sports center built to a scale of $1:250$, the Olympic swimming pool occupies a surface area of $20\text{ cm}^2$ and has a capacity (volume) of $160\text{ cm}^3$.

• a) Calculate the actual surface area of the pool in square meters ($m^2$).

• b) Calculate the actual capacity of the pool in liters ($L$). (Remember that $1\text{ dm}^3 = 1\text{ L}$).

• c) If we want to make another smaller model where the surface area of the pool is only $5\text{ cm}^2$, what scale should we use?

A drone takes an aerial photograph of a car $4.5\text{ meters}$ in length. On the phone screen, which shows the photo at full size, the car measures $1.8\text{ cm}$.

• a) What is the scale of the image on the screen?

• b) In the same photo, there is a rectangular garden that measures $5\text{ cm}$ long and $3\text{ cm}$ wide on the screen. What is the actual area of the garden in square meters?

• c) If the drone flies to twice the height, how long will the car measure on the phone screen if the camera zoom is unchanged?

We have two similar containers for popcorn. The small container has a height of $15\text{ cm}$ and the large one a height of $25\text{ cm}$. If the cost of the cardboard to make the small container is $\text{RD\$}\ 10.80$, and the price of the cardboard is proportional to its surface area:

• a) What will the cost of the cardboard needed to make the large container be?

• b) If the small container holds $1.2\text{ kg}$ of popcorn, how many kilograms of popcorn will fit in the large container? (Assume both are filled to the brim).

An architect presents the plan of an apartment at a scale of $1:50$. In the living room, which is perfectly rectangular, they have drawn a rug measuring $6\text{ cm}$ long and $4\text{ cm}$ wide. The client asks for the actual rug to be 20% longer and 20% wider.

• a) What are the actual dimensions of the rug requested by the client?

• b) What is the scale factor between the drawn rug and the final actual rug?

• c) By what percentage has the area of the actual rug increased compared to what was originally planned on the plan?

Two engine parts are similar. Part $A$ weighs $2.4\text{ kg}$ and is made of a material costing $15\text{ \text{RD\$}\ /kg}$. Part $B$ is a reduction of Part $A$ with a scale factor of $k = 0.5$.

• a) If both parts are made of the same dense material, how much will Part $B$ weigh?

• b) What will the material cost of Part $B$ be?

• c) If the surface of Part $A$ must be covered with a rustproofing coating costing $\text{RD\$}\ 2$ per $10\text{ cm}^2$ and the treatment for Part $A$ costs $\text{RD\$}\ 48$, how much will the treatment for Part $B$ cost?

A map has a linear scale where a segment of $4\text{ cm}$ represents $2\text{ km}$ in reality. On this map, a nature reserve is in the shape of an equilateral triangle with side $6\text{ cm}$.

• a) Calculate the ratio scale of the map.

• b) Calculate the actual perimeter of the reserve in kilometers.

• c) If on a new map the same reserve has an area 4 times larger than on the first map, what is the scale of the new map?

A courier company uses similar cubical boxes. The "Standard" box has an edge length of $20\text{ cm}$. The "Maxi" box has a total surface area that is exactly double that of the "Standard" box.

• a) Calculate the scale factor between the "Maxi" box and the "Standard" box (express the result to two decimal places).

• b) What is the edge length of the "Maxi" box?

• c) If the "Standard" box can hold $8\text{ liters}$ of sand, how many liters can the "Maxi" box hold?