The problem in the image involves solving for the variable $x$ using similar triangles. Let's break it down step by step:

1. Observing the triangles

From the image:

• The two triangles are similar by angle-angle similarity (corresponding angles are equal).
• This means their sides are proportional.

Labeling sides:

• The larger triangle has sides 12 (vertical), 2x - 3 (horizontal), and hypotenuse.
• The smaller triangle has sides 3 (vertical), $x - 5$ (horizontal), and hypotenuse.

2. Setting up the proportion

Using the property of similar triangles: $\frac{\text{Vertical side of large triangle}}{\text{Vertical side of small triangle}} = \frac{\text{Horizontal side of large triangle}}{\text{Horizontal side of small triangle}}$

Substituting values: $\frac{12}{3} = \frac{2x - 3}{x - 5}$

3. Simplifying the proportion

Simplify $\frac{12}{3}$: $4 = \frac{2x - 3}{x - 5}$

4. Solving for $x$

Cross-multiply: $4(x - 5) = 2x - 3$

Expand the terms: $4x - 20 = 2x - 3$

Simplify: $4x - 2x = -3 + 20$

$2x = 17$

Divide by 2: $x = 8.5$

Final Answer:

$x = 8.5$

Would you like a deeper explanation or verification for the solution?

Related Questions:

1. What are the criteria for triangle similarity, and how are they applied?
2. How does the proportionality rule work in similar triangles?
3. Can you derive $x$ using another pair of corresponding sides in these triangles?
4. What happens if one side of the triangle is not parallel (affecting similarity)?
5. How can the problem change if the given values for sides differ?

Tip:

Always check if triangles are similar before using proportions to solve problems!