By similarity, we can set up the proportion:

\(\displaystyle \frac{BX}{AD} = \frac{BC}{AB}\)

Substitute: 

\(\displaystyle AB=12, AD=BC=5\)

\(\displaystyle \frac{BX}{5} = \frac{5}{12}\)

\(\displaystyle BX = \frac{5}{12} \cdot 5 =\frac{25}{12} = 2 \frac{1}{12}\)

For polygons to be similar, side lengths must be in proportion.

Case 1:

\(\displaystyle x\) and 100 in the first rectangle correspond to \(\displaystyle x\) and 400 in the second, respectively.

The resulting proportion would be:

\(\displaystyle \frac{x}{400} = \frac{x}{100}\)

\(\displaystyle 100x=400x\)

\(\displaystyle 300x=0\)

\(\displaystyle x=0\)

This is impossible since \(\displaystyle x\) must be a positive side length.

Case 2:

\(\displaystyle x\) and 100 in the first rectangle correspond to 400 and \(\displaystyle x\) in the second, respectively.

The correct proportion statement must be:

\(\displaystyle \frac{x}{400} = \frac{100}{x}\)

Cross multiply to solve for \(\displaystyle x\):

\(\displaystyle x\cdot x = 400 \cdot 100\)

\(\displaystyle x^{2} = 40,000\)

\(\displaystyle x = \sqrt{40,000} = 200\)

200 cm is the only possible solution.

All sides being equal is a condition for congruency, not similarity. Similarity focuses on the ratio between rectangles and not on the equivalency of all sides. As for the statement regarding the equal angles, all rectangles regardless of similarity or congruency have four 90 degree angles. 

For two rectangles to be similar, their sides have to be proportional (form equal ratios). The ratio of the two longer sides should equal the ratio of the two shorter sides.

\(\displaystyle \frac{99}{45}=\frac{44}{x}\)

However, the left ratio in our proportion reduces.

\(\displaystyle \frac{11}{5}=\frac{44}{x}\)

We can then solve by cross multiplying.

\(\displaystyle 11* x=44*5\)

\(\displaystyle 11x=220\)

We then solve by dividing.

\(\displaystyle x=20\)

For two rectangles to be similar, their sides must be in the same ratio.

This problem can be solved using ratios and cross multiplication. 

\(\displaystyle \frac{w_{left}}{l_{left}}=\frac{w_{right}}{l_{right}}\)

Let's denote the unknown width of the right rectangle as x. 

\(\displaystyle w_{right}=x\)

\(\displaystyle \frac{{\color{Magenta} 6}}{{\color{Green} 10}}=\frac{{\color{Green} x}}{{\color{Magenta} 7}}\)

\(\displaystyle 10x=7\cdot 6\)

\(\displaystyle 10x=42\)

\(\displaystyle x=\frac{42}{10}\)

\(\displaystyle {\color{Blue} x=4.2}\)

The goal is to solve for the height of the second rectangle. 

Similar rectangles function on proportionality - that is, the ratios of the sides between two rectangles will be the same. In order to determine the height, we will be using this concept of ratios through solving for variables from the area.

First, it's helpful to achieve full dimensions for the first rectangle.

It is given that its base length is 5, and it has an area of 20. 
\(\displaystyle Area = b\cdot h\)
\(\displaystyle 20=5\cdot h\)
\(\displaystyle {\color{Blue} h=4}\)

This means the first rectangle has the dimensions 5x4. 

Now, we may utilize the concept of ratios for similarity. The side lengths of the first rectangle is 5x4, so the second recatangle must have sides that are proportional to the first's.

\(\displaystyle \frac{base_1}{height_1}=\frac{base_2}{height_2}\)

We have the information for the first rectangle, so the data may be substituted in.

\(\displaystyle \frac{4}{5}=\frac{b_2}{h_2}\)

\(\displaystyle 5b_2=4h_2\) 

\(\displaystyle b_2=\frac{4}{5}h_2\) is the ratio factor that will be used to solve for the height of the second rectangle. This may be substituted into the area formula for the second rectangle. 

\(\displaystyle Area= b \cdot h\)

\(\displaystyle 80=\left(\frac{4}{5}h\right)\cdot h\)

\(\displaystyle \frac{5}{4} \cdot 80= h \cdot h\)

\(\displaystyle 100=h^2\)

\(\displaystyle {\color{Blue} 10=h}\)

Therefore, the height of the second rectangle is 10.

The goal of this problem is to figure out what base length of the second rectangle will make it similar to the first rectangle.

Similar rectangles function on proportionality - that is, the ratios of the sides between two rectangles will be the same. In order to determine the base, we will be using this concept of ratios through solving for variables from the perimeter.

First, all the dimensions of the first rectangle must be calculated.

This can be accomplished through using the perimeter equation: \(\displaystyle P=2h+2b\)

\(\displaystyle 30=2(10)+2b\)

\(\displaystyle 30=20+2b\)

\(\displaystyle 10=2b\)

\(\displaystyle {\color{Blue} 5=b}\)

This means the dimensions of the first rectangle are 10x5. We will use this information for the ratios to calculate dimensions that would yield the second rectangle similar because of proprotions. 

\(\displaystyle \frac{b_1}{h_1}=\frac{b_2}{h_2}\)

\(\displaystyle \frac{5}{10}=\frac{b_2}{h_2}\)

\(\displaystyle 10b_2=5h_2\)

\(\displaystyle h_2=2b_2\) is the ratio factor we will use to solve for the base of the second rectangle. 

This will require revisiting the perimeter equation for the second rectangle.

\(\displaystyle P=2h+2b\)

\(\displaystyle 50=2(2b)+2b\)

\(\displaystyle 50=4b+2b\)

\(\displaystyle 50=6b\)

\(\displaystyle \frac{50}{6}=b\)

\(\displaystyle {\color{Blue} b=8.3}\)

Two rectangles are similar if their length and width form the same ratio. The small tile has a width of \(\displaystyle 17\;cm\) and a width of \(\displaystyle 34\;cm\), providing us with the following ratio:

\(\displaystyle \frac{34\;cm}{17\;cm}= 2\)

Since the length of similar triangles is twice their respective width, the length of the large tile can be determined as such:

\(\displaystyle \\length=2\cdot\ width\\length=28\;cm\cdot2\\length=56\;cm\)

To determine if these rectangles are similar, set up a proportion:

\(\displaystyle \frac{13}{15} = \frac{4}{6}\) This proportion compares the ratio between the long sides in each rectangle to the ratio of the short sides in each rectangle. If they are the same, cross-multiplying will produce a true statement, and the rectangles are similar:

\(\displaystyle 78 \neq 60\)

These rectangles aren't similar.

To determine if the rectangles are similar, set up a proportion comparing the short sides and the long sides from each rectangle: 

\(\displaystyle \frac{3}{7.5 } = \frac{10}{25 }\) cross-multiply

\(\displaystyle 75 = 75\) since that's true, the rectangles are similar.

To find the scale factor, either divide 25 by 10 or 7.5 by 3. Either way you will get 2.5.