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To solve for \( PS \), \( SR \), and \( QS \) in the given triangle:

We have:
- \( PQ = 6 \)
- \( QR = 8 \)
- \( QS \) is perpendicular to \( PR \), making triangles \( \triangle PQS \), \( \triangle QSR \), and \( \triangle PQR \) all right-angled triangles.

### Step 1: Find \( PR \) (the length of the hypotenuse)
Using the Pythagorean theorem in \( \triangle PQR \):
\[
PR = \sqrt{PQ^2 + QR^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]

### Step 2: Use the property of similar triangles
Since \( QS \) is perpendicular to \( PR \), triangles \( \triangle PQS \), \( \triangle QSR \), and \( \triangle PQR \) are similar. Using the ratios of the sides for the similar triangles, we have the following relationships:
\[
\frac{PS}{PQ} = \frac{QS}{PS} = \frac{SR}{QR}
\]

### Step 3: Find \( PS \), \( SR \), and \( QS \)
I'll calculate these values next.We have calculated that the length of \( PR \) (the hypotenuse) is \( 10 \).

Next, we will use the fact that the triangles \( \triangle PQS \), \( \triangle QSR \), and \( \triangle PQR \) are similar to establish the relationships between the sides and solve for \( PS \), \( SR \), and \( QS \).It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like me to continue explaining the next steps to solve this manually? Here are five related questions for practice:

1. How do you apply the Pythagorean theorem to solve for unknown sides of a right triangle?
2. What are the properties of similar triangles in terms of side ratios?
3. How would you find the area of the right-angled triangle \( \triangle PQR \)?
4. How does the perpendicular line \( QS \) divide triangle \( PQR \) into two smaller right triangles?
5. Can you derive the formula for finding the length of the altitude \( QS \) from vertex \( Q \)?

**Tip:** When triangles are similar, the ratios of corresponding sides are equal, which helps in solving for unknown lengths.

In right-angled ΔPQR, QS is a perpendicular dropped on the hypotenuse PR from vertex Q. Given PQ = 6 and QR = 8, find PS, SR, and QS.

This problem involves finding the lengths of segments PS, SR, and QS in a right-angled triangle ΔPQR using the Pythagorean theorem and properties of similar triangles. Given PQ = 6 and QR = 8, the hypotenuse PR is calculated, and then segment lengths are determined.

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