Concepts of Similarity

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.

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.

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.

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?

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?

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.

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.

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.

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 meters.

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?

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 meters.
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).

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.

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 liters. 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?

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 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 meters.
2. b) If the real building is to be covered with glass panels costing $7\,200$ pesos per square meter, 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 $180 \text{ cm}^2$.

To measure the height of an electricity pylon, a technician observes that at a particular moment the pylon casts a shadow of $18\text{ meters}$. At the same time, they place a vertical stick $1.5\text{ meters}$ high, which casts 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.

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.

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.

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 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 $0.25$ pesos per $\text{cm}^3$, calculate the price difference between the contents of both bottles.

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 centimeters.
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?