The problem states that two triangles are similar. The dimensions of the sides of the triangles are given in centimeters. a. We need to find the value of $x$. b. We need to find the scale factor from the smaller triangle to the larger triangle.

1. Problem Description

The problem states that two triangles are similar. The dimensions of the sides of the triangles are given in centimeters.

a. We need to find the value of $x$.

b. We need to find the scale factor from the smaller triangle to the larger triangle.

2. Solution Steps

a. Since the triangles are similar, the corresponding sides are proportional. The side of length 8 in the smaller triangle corresponds to the side of length 9 in the larger triangle. The side of length 10 in the smaller triangle corresponds to the side of length 12 in the larger triangle. The side of length $x$ in the smaller triangle corresponds to the unnamed side of the larger triangle. We can set up the proportion:

$\frac{x}{9} = \frac{8}{9} = \frac{10}{12}$

We can also set up the proportion:

$\frac{x}{9} = \frac{8}{9}$ or $\frac{x}{9} = \frac{10}{12}$

From $\frac{8}{9} = \frac{10}{12}$, we can calculate the scaling factor:

$\frac{10}{12} = \frac{5}{6}$

Then we can calculate $x$ by using $\frac{x}{9} = \frac{8}{9}$:

$x = 9 \cdot \frac{8}{9} = 8$

Using $\frac{x}{9} = \frac{10}{12}$, we can also calculate $x$.

$x = 9 \cdot \frac{10}{12} = \frac{90}{12} = \frac{30}{4} = \frac{15}{2} = 7.5$

Since the two given sides are corresponding, the sides must be in the correct order. The side of length 8 corresponds to 9, and the side of length 10 corresponds to

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2. Therefore, we have:

$\frac{x}{9} = \frac{8}{9} = \frac{10}{12}$ must hold true.

Since the equality $\frac{8}{9} = \frac{10}{12}$ doesn't hold, the correspondence defined as such is incorrect.

Looking at the diagram, we can see that $x$ corresponds to the side of length 9 since they are both opposite the angle marked with a single line. The side of length 8 corresponds to the side of length 12 since they are both opposite the angle marked with two lines. Therefore, the ratio is:

$\frac{x}{9} = \frac{8}{12}$

$x = 9 \cdot \frac{8}{12} = 9 \cdot \frac{2}{3} = 3 \cdot 2 = 6$

b. The scale factor from the smaller triangle to the larger triangle is the ratio of the corresponding sides. Using the sides with lengths 8 and 12, the scale factor is:

$\frac{12}{8} = \frac{3}{2} = 1.5$

Using the sides with lengths 6 and 9, the scale factor is:

$\frac{9}{6} = \frac{3}{2} = 1.5$

Using the sides with lengths 10 and an unnamed side, we can calculate this as 1.5 times 10, which will be

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5. Thus, the correspondence is $10 \to 15$.

$\frac{15}{10} = \frac{3}{2} = 1.5$

3. Final Answer

a. $x = 6$

b. $\frac{3}{2}$ or $1.5$