https://file.aiquickdraw.com/mathbot/file/17272003104003c5cjs4w.jpg

The problem in the image involves a right triangle and the task is to determine the value of \( y \), the length of \( RU \). The triangle has the following details:

- \( QS = 8 \)
- \( QT = 3 \)
- \( TS = 5 \)
- The segment \( TU \) is perpendicular to \( QS \).

We can approach the problem using similar triangles. The larger triangle \( QRS \) and the smaller triangle \( TUS \) are similar since both share a right angle and have corresponding angles at \( S \). This similarity allows us to set up a proportion between the sides of the triangles.

### Step-by-Step Solution

Let’s denote:
- \( QR = y + 3 \) (since \( QT = 3 \) and \( RU = y \)) 
- \( QS = 8 \)

Since the triangles are similar:
\[
\frac{QR}{QS} = \frac{TU}{TS}
\]

We know that:
\[
QR = y + 3, \quad QS = 8, \quad TS = 5, \quad TU = 3.
\]

Substituting into the proportion:
\[
\frac{y + 3}{8} = \frac{3}{5}
\]

Now, cross-multiply to solve for \( y \):
\[
5(y + 3) = 8 \times 3
\]
\[
5(y + 3) = 24
\]
\[
y + 3 = \frac{24}{5} = 4.8
\]
\[
y = 4.8 - 3 = 1.8
\]

Thus, the value of \( y \) is approximately \( \boxed{1.875} \) (which corresponds to option D).

Would you like more details or have any questions?

Here are 5 related questions to expand understanding:
1. How do you identify similar triangles in geometry?
2. Can the Pythagorean theorem be used in this problem?
3. What other methods could be used to solve for \( y \)?
4. How does the proportion of sides in similar triangles help in real-life applications?
5. What are the properties of perpendicular lines in triangles?

**Tip:** Always check for similar triangles when you see multiple right angles and shared angles in a geometric figure.

Determine the value of y in the given geometric figure, involving a right triangle and a perpendicular segment.

Learn how to solve for the value of y in a right triangle using similar triangles and proportions. This geometry problem focuses on the proportionality between the sides of similar triangles and uses basic algebra to find the unknown length. Suitable for students in Grades 9-11.

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