Apply the Pythagorean Theorem to find the length between (2, 5) and (7, 3) round your answer to the nearest hundreth if nesscacary.

To find the distance between the points (2, 5) and (7, 3) using the Pythagorean theorem, we can treat the distance as the hypotenuse of a right triangle where the legs are the differences in the coordinates.

1. Calculate the difference in the x-coordinates:
\( \Delta x = 7 - 2 = 5 \)

2. Calculate the difference in the y-coordinates:
\( \Delta y = 3 - 5 = -2 \)
(The absolute value is \( 2 \), but the sign doesn't affect the distance.)

3. Use the Pythagorean theorem:
\( d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \)
\( d = \sqrt{(5)^2 + (-2)^2} \)
\( d = \sqrt{25 + 4} \)
\( d = \sqrt{29} \)

4. Calculate \( \sqrt{29} \):
\( \sqrt{29} \approx 5.385 \)

Rounding to the nearest hundredth gives us:
**5.39**

Thus, the length between (2, 5) and (7, 3) is **5.39**.

you sure??

Let’s double-check the calculations for the distance between the points (2, 5) and (7, 3).

1. **Identify the coordinates:**
- Point A: (2, 5)
- Point B: (7, 3)

2. **Calculate the differences in coordinates:**
- \( \Delta x = x_2 - x_1 = 7 - 2 = 5 \)
- \( \Delta y = y_2 - y_1 = 3 - 5 = -2 \)

3. **Apply the Pythagorean theorem:**
- The distance \( d \) is given by:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
- Plugging in the values:
\[
d = \sqrt{(5)^2 + (-2)^2}
\]
- Calculate the squares:
\[
d = \sqrt{25 + 4}
\]
- Combine the results:
\[
d = \sqrt{29}
\]

4. **Calculate \( \sqrt{29} \):**
- Using a calculator, \( \sqrt{29} \approx 5.385164807 \).

5. **Round to the nearest hundredth:**
- This gives us approximately **5.39**.

So, yes, the distance between the points (2, 5) and (7, 3) is indeed **5.39**. Thank you for your patience, and feel free to ask if you have any further questions!

can you do that again but with these numbers? (-1, 6) and (5, -2)

10
100
9.22
14

Sure! Let's find the distance between the points \((-1, 6)\) and \((5, -2)\) using the Pythagorean theorem.

1. **Identify the coordinates:**
- Point A: \((-1, 6)\)
- Point B: \((5, -2)\)

2. **Calculate the differences in coordinates:**
- \( \Delta x = x_2 - x_1 = 5 - (-1) = 5 + 1 = 6 \)
- \( \Delta y = y_2 - y_1 = -2 - 6 = -8 \)

3. **Apply the Pythagorean theorem:**
- The distance \( d \) is given by:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
- Plugging in the values:
\[
d = \sqrt{(6)^2 + (-8)^2}
\]
- Calculate the squares:
\[
d = \sqrt{36 + 64}
\]
- Combine the results:
\[
d = \sqrt{100}
\]

4. **Calculate \( \sqrt{100} \):**
- This equals \( 10 \).

So, the distance between the points \((-1, 6)\) and \((5, -2)\) is **10**.

can you double check?

Sure! Let's carefully re-evaluate the calculation for the distance between the points \((-1, 6)\) and \((5, -2)\) step by step.

1. **Identify the points:**
- Point A: \((-1, 6)\)
- Point B: \((5, -2)\)

2. **Calculate the differences in coordinates:**
- \( \Delta x = x_2 - x_1 = 5 - (-1) = 5 + 1 = 6 \)
- \( \Delta y = y_2 - y_1 = -2 - 6 = -8 \)

3. **Calculate the distance using the Pythagorean theorem:**
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
Plugging in the values:
\[
d = \sqrt{(6)^2 + (-8)^2}
\]
Calculating the squares:
\[
(6)^2 = 36
\]
\[
(-8)^2 = 64
\]
Now combine these results:
\[
d = \sqrt{36 + 64}
\]
\[
d = \sqrt{100}
\]

4. **Final calculation:**
- \( \sqrt{100} = 10 \)

So, the distance between the points \((-1, 6)\) and \((5, -2)\) is indeed **10**. The calculations confirm the result. Thank you for your patience, and if you have any more questions or need further assistance, feel free to ask!

can you do it again with these numbers? (5, 5) and (2, -3)

73
11
8.54
7.62

Let's find the distance between the points \((5, 5)\) and \((2, -3)\) using the Pythagorean theorem, carefully applying each step.

1. **Identify the points:**
- Point A: \((5, 5)\)
- Point B: \((2, -3)\)

2. **Calculate the differences in coordinates:**
- \( \Delta x = x_2 - x_1 = 2 - 5 = -3 \)
- \( \Delta y = y_2 - y_1 = -3 - 5 = -8 \)

3. **Calculate the distance using the Pythagorean theorem:**
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
Plugging in the values:
\[
d = \sqrt{(-3)^2 + (-8)^2}
\]
Calculating the squares:
\[
(-3)^2 = 9
\]
\[
(-8)^2 = 64
\]
Now combine these results:
\[
d = \sqrt{9 + 64}
\]
\[
d = \sqrt{73}
\]

4. **Final calculation:**
- Using a calculator, \( \sqrt{73} \approx 8.544 \).

5. **Round to the nearest hundredth:**
- This gives us approximately **8.54**.

So, the distance between the points \((5, 5)\) and \((2, -3)\) is **8.54**.

alr now cna you double check every problem you solved for me?

Absolutely! Let's go through each problem one by one, double-checking the calculations.

### Problem 1: Distance between (2, 5) and (7, 3)

1. **Coordinates:**
- Point A: (2, 5)
- Point B: (7, 3)

2. **Calculate differences:**
- \( \Delta x = 7 - 2 = 5 \)
- \( \Delta y = 3 - 5 = -2 \)

3. **Apply Pythagorean theorem:**
\[
d = \sqrt{(5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}
\]

4. **Calculate \( \sqrt{29} \):**
- \( \sqrt{29} \approx 5.385 \)

5. **Round to the nearest hundredth:**
- \( d \approx 5.39 \)

**This is correct: The distance is 5.39.**

### Problem 2: Distance between (-1, 6) and (5, -2)

1. **Coordinates:**
- Point A: (-1, 6)
- Point B: (5, -2)

2. **Calculate differences:**
- \( \Delta x = 5 - (-1) = 5 + 1 = 6 \)
- \( \Delta y = -2 - 6 = -8 \)

3. **Apply Pythagorean theorem:**
\[
d = \sqrt{(6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100}
\]

4. **Calculate \( \sqrt{100} \):**
- \( \sqrt{100} = 10 \)

**This is correct: The distance is 10.**

### Problem 3: Distance between (5, 5) and (2, -3)

1. **Coordinates:**
- Point A: (5, 5)
- Point B: (2, -3)

2. **Calculate differences:**
- \( \Delta x = 2 - 5 = -3 \)
- \( \Delta y = -3 - 5 = -8 \)

3. **Apply Pythagorean theorem:**
\[
d = \sqrt{(-3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73}
\]

4. **Calculate \( \sqrt{73} \):**
- Using a calculator, \( \sqrt{73} \approx 8.544 \).

5. **Round to the nearest hundredth:**
- \( d \approx 8.54 \)

**This is correct: The distance is 8.54.**

### Summary of Results:
1. Distance between (2, 5) and (7, 3) = **5.39**
2. Distance between (-1, 6) and (5, -2) = **10**
3. Distance between (5, 5) and (2, -3) = **8.54**

All calculations have been verified and are correct. If you have any further questions or if there's anything else you'd like to check, feel free to ask!