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.

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?

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.

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 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.

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 litres. 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?

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 $14\text{ cm}$.