Students identify and correct mistakes in calculating distance between two points using the Pythagorean theorem.

Problem Analysis:

The task involves identifying and correcting errors made by two students, Delilah and Tucker, when calculating the distance between two points using the Pythagorean Theorem. The Pythagorean Theorem states:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

#### Step 1: Analyze Delilah's Work
Delilah's calculations are as follows:
- \(x\)-coordinates: \(4\) and \(1\)
- \(y\)-coordinates: \(3\) and \(2\)

Delilah's steps:
1. Calculate \(x\)-difference: \(|4 - 1| = 3\)
2. Calculate \(y\)-difference: \(|3 - 2| = 1\)
3. Square the differences:
- \(3^2 = 9\)
- \(1^2 = 1\)
4. Add the squares: \(9 + 1 = 10\)
5. Take the square root: \(\sqrt{10} \approx 3.16\)

However, Delilah concludes the distance as 3.46 units, which is incorrect.

#### Error in Delilah's Work:
Delilah correctly calculated the differences and squared them, but she made an error in the final step. She incorrectly concluded that the distance was 3.46 units instead of the correct value of \(\sqrt{10} \approx 3.16\).

#### Correct Solution for Delilah:
Using the Pythagorean Theorem:
\[
d = \sqrt{(4 - 1)^2 + (3 - 2)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.16
\]

#### Final Answer for Delilah:
- Error: Incorrectly stated the final distance as 3.46 units.
- Correct Distance: \(\boxed{3.16}\)

---

#### Step 2: Analyze Tucker's Work
Tucker's calculations are as follows:
- \(x\)-coordinates: \(4\) and \(1\)
- \(y\)-coordinates: \(3\) and \(2\)

Tucker's steps:
1. Calculate \(x\)-difference: \(|4 - 1| = 3\)
2. Calculate \(y\)-difference: \(|3 - 2| = 1\)
3. Square the differences:
- \(3^2 = 9\)
- \(1^2 = 1\)
4. Add the squares: \(9 + 1 = 10\)
5. Take the square root: \(\sqrt{10} \approx 3.16\)

However, Tucker concludes the distance as 28 units, which is incorrect.

#### Error in Tucker's Work:
Tucker correctly calculated the differences and squared them, but he made a significant error in the final step. Instead of taking the square root of 10, he incorrectly concluded that the distance was 28 units.

#### Correct Solution for Tucker:
Using the Pythagorean Theorem:
\[
d = \sqrt{(4 - 1)^2 + (3 - 2)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.16
\]

#### Final Answer for Tucker:
- Error: Incorrectly stated the final distance as 28 units.
- Correct Distance: \(\boxed{3.16}\)

---

Summary of Solutions:

1. Delilah's Error: Incorrectly stated the final distance as 3.46 units.
- Correct Distance: \(\boxed{3.16}\)

2. Tucker's Error: Incorrectly stated the final distance as 28 units.
- Correct Distance: \(\boxed{3.16}\)