Exercises: Concepts

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

Calculate the scale factor $k$ that converts 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 measurement of all its sides, how much will the angles of the new triangle measure? Explain what happens to the angles of two figures when they are similar.

On his cell phone screen, Marcos sees a map at 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 the streets change on the map so that they can fit on the screen.

A logo has the shape of a regular pentagon with a side of $3 \text{ cm}$. For a store facade, a similar logo is manufactured with a scale factor of $k = 15$. How much will the side of the new logo measure?

Look at the following pairs of rectangles. State, without performing complex calculations, which of them 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 that exists between the scale factor and perimeters.

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

An illustrator is designing two versions of the same rectangular logo. The original logo has dimensions of $12\text{ cm}$ long and $8\text{ cm}$ wide. 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 perimeter of the small logo is $40\text{ cm}$, what will the perimeter of the large one be? Use the relationship between perimeters of similar figures to check it.

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

Two triangles, $ABC$ and $A'B'C'$, are in Thales's 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 and the smaller triangle?

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

3. Check that the quotient between 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 $12\text{ in}$ long by $8\text{ in}$ wide.

1. Calculate the real dimensions of the room in feet.

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

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

To measure the height of a high-voltage pylon, a technician observes that at a certain moment the pylon projects a shadow of $18\text{ meters}$. At the same time, they place a vertical stick $1,5\text{ meters}$ high that projects a shadow of $2,25\text{ meters}$.

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 if the following pairs of triangles are similar and state which similarity criterion you have 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 one, the sides that form that angle measure $4\text{ cm}$ and $7\text{ cm}$. In the second one, they measure $8\text{ cm}$ and $14\text{ cm}$.

In a right triangle, the height on the hypotenuse divides it into two segments (projections of the legs) that measure $3,6\text{ cm}$ and $10\text{ cm}$.

1. Calculate the height of the triangle using the Altitude Theorem.

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

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

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

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

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

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

A perfume company launches two formats of its flagship bottle, which have the shape of similar square 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 glass surface area is exactly one-ninth of the surface area of the "Large" bottle. If the cost of the perfume is $\$0,25$ for every $\text{cm}^3$, calculate the difference in price 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 this, he places a spotlight on the ground at a distance $d$ from the sculpture. If he wants the shadow projected on the wall to be exactly $8 \text{ m}$ high, at what distance from the sculpture should he place the spotlight? Use Thales's Theorem to justify your answer and draw the diagram of the similar triangles.

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

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

1. a) Calculate the actual surface area of the kitchen in square feet.
2. b) What is the minimum number of whole tiles needed to cover the actual kitchen floor? (Keep in mind that offcuts can be reused, but tiles can only be bought whole).

Two regular pentagons are similar. The perimeter 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 perimeter of the second pentagon.
3. c) If the apothem of the first pentagon measures $10,32 \text{ cm}$, how much 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 $2,54 \text{ in}$ and that the "inches" of a television refer to the measurement of its diagonal:

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

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

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