Similarity 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.

Cian 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.

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.

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?

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, Cian 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 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?

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).

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?

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.

The interior courtyard of a cultural centre in your county 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?

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).

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 $14\text{ cm}$.

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.

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.

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.