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The hole locations for the steel plate shown are to be programmed into a numerically controlled jig borer. The turntable on the borer requires that the holes be located by giving the angle θ of each hole and the distance r to each hole, as measured from the hole in the bottom left corner. Convert the dimensions of third and fifth holes.

To convert the dimensions of the third and fifth holes to polar coordinates \((r, \theta)\), we’ll calculate the distance \( r \) from the origin (bottom-left hole) to each hole and the angle \( \theta \) relative to the horizontal. Here’s how:

### 1. Third Hole Conversion

The third hole is located:
- 3.8 cm + 3.1 cm = 6.9 cm from the left edge horizontally
- 8.6 cm vertically from the bottom edge

#### Calculations:
- **Distance \( r \):**  
  \( r = \sqrt{(6.9)^2 + (8.6)^2} \)
- **Angle \( \theta \):**  
  \( \theta = \tan^{-1}\left(\frac{8.6}{6.9}\right) \)

### 2. Fifth Hole Conversion

The fifth hole is located:
- 18 cm from the left edge horizontally (same as total width)
- 3.4 cm + 2.7 cm = 6.1 cm vertically from the bottom edge

#### Calculations:
- **Distance \( r \):**  
  \( r = \sqrt{(18)^2 + (6.1)^2} \)
- **Angle \( \theta \):**  
  \( \theta = \tan^{-1}\left(\frac{6.1}{18}\right) \)

Let me compute these values.The polar coordinates for the third and fifth holes are as follows:

1. **Third Hole**  
   - Distance \( r = 11.03 \) cm
   - Angle \( \theta = 51.26^\circ \)

2. **Fifth Hole**  
   - Distance \( r = 19.01 \) cm
   - Angle \( \theta = 18.72^\circ \)

Would you like further details on the calculations or have additional questions?

Here are some related questions for further exploration:
1. How are polar coordinates beneficial in programming CNC machines?
2. What would be the polar coordinates for the other holes?
3. How does one convert from polar to Cartesian coordinates?
4. What trigonometric functions are commonly used in polar coordinate conversions?
5. How could errors in angle measurements affect CNC machining?

**Tip:** When working with polar coordinates, always ensure the angle \( \theta \) is measured consistently from the same reference direction.

Learn how to convert Cartesian coordinates to polar coordinates for CNC machining. This example covers calculations for the third and fifth hole locations on a steel plate, using the Pythagorean theorem and trigonometric functions. Suitable for high school students studying geometry and trigonometry.

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