https://file.aiquickdraw.com/mathbot/file/1730722155879nv5yzw76.jpg

To solve this problem, we need to analyze the given shape \( PQRSTU \) with labeled side lengths, all of which are expressed in terms of \( a \). Since the shape has only right angles, we can determine missing sides by ensuring the lengths align with the overall geometry of the figure.

### Step 1: Label Known Sides
From the image, we have:
1. **\( PQ \)** (vertical side) = \( 8a + 3 \)
2. **\( UT \)** (vertical side) = \( 2a + 3 \)
3. **\( SR \)** (vertical side) = \( a \)
4. **\( PU \)** (horizontal side) = \( 5a - 1 \)

### Step 2: Determine Missing Side Lengths

#### Finding \( QR \) (bottom horizontal side)
Since \( QR \) extends horizontally from \( Q \) to \( R \), we can express \( QR \) as the sum of \( PU \) and \( TS \):
\[
QR = PU + TS
\]

#### Finding \( TS \) (horizontal side)
From the structure, \( TS \) can be calculated by subtracting \( UT \) from \( PQ \) (since \( PQ \) is the full vertical height and \( UT \) is a segment of it):
\[
TS = PQ - UT = (8a + 3) - (2a + 3) = 6a
\]

#### Substitute to find \( QR \):
\[
QR = (5a - 1) + 6a = 11a - 1
\]

### Summary of Answers
1. **\( TS = 6a \)**
2. **\( QR = 11a - 1 \)**

Would you like further details, or do you have questions on this solution?

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### Further Related Questions
1. How would we find the perimeter of this shape?
2. What would be the area of this shape in terms of \( a \)?
3. If \( a = 2 \), what would be the exact lengths of all sides?
4. How would the shape change if all side lengths doubled?
5. Can you calculate the internal angles of this shape?

### Tip
In composite shapes, break down complex sides into smaller sections by identifying which lengths add or subtract based on geometry.

The diagram shows the shape PQRSTU. All of the corners are right angles. The lengths of four of the sides have been given in terms of a.

Solve the geometry problem involving the shape PQRSTU with all right angles. Learn how to calculate the lengths of sides using algebraic expressions in terms of a. Suitable for Grades 7-9.

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