The problem in the image asks to find the length $x$ in a pair of right triangles that appear to be similar.

Given dataThe problem in the image asks to find the length $x$ in a pair of right triangles that appear to be similar.

Given

• One triangle has sides of lengths 2 (adjacent) and 4 (hypotenuse).
• The larger triangle has sides 5 (adjacent) and $x$ (hypotenuse).

Since the triangles are similar, the ratios of corresponding sides are equal. Therefore, we can set up the following proportion using the sides of the triangles:

$\frac{2}{4} = \frac{5}{x}$

Now, solve for $x$:

$\frac{2}{4} = \frac{5}{x} \implies \frac{1}{2} = \frac{5}{x}$

Cross-multiply:

$1 \cdot x = 2 \cdot 5$

$x = 10$

Thus, the length of $x$ is 10.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine if two triangles are similar?
2. What is the relationship between corresponding angles in similar triangles?
3. Can you use the Pythagorean Theorem to verify the side lengths of a right triangle?
4. How do proportionality and scaling work in similar triangles?
5. How would the solution change if you were given the opposite sides instead of adjacent sides?

Tip: When dealing with similar triangles, always check if their angles are equal; this guarantees that the sides are proportional.