Question

Find the length of the red line segment in the coordinate plane. (-2,1) and (4,3)
Which of the following statements are true?

the length is 40 units: find the sum of x and y coordinates and square root the answer
the length is square root 260: find the difference of the x coordinates and the difference of the y coordinates and subtract those differences
the length is square root 40 units:
the length is 130

Answer 1

To find the length of the line segment connecting the points \((-2, 1)\) and \((4, 3)\), we can use the distance formula, which is given by:

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

Where:
- \((x_1, y_1) = (-2, 1)\)
- \((x_2, y_2) = (4, 3)\)

Now we can calculate:

1. **Calculate the differences:**
\[
x_2 - x_1 = 4 - (-2) = 4 + 2 = 6
\]
\[
y_2 - y_1 = 3 - 1 = 2
\]

2. **Squares of the differences:**
\[
(x_2 - x_1)^2 = 6^2 = 36
\]
\[
(y_2 - y_1)^2 = 2^2 = 4
\]

3. **Add the squares:**
\[
36 + 4 = 40
\]

4. **Take the square root:**
\[
d = \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}
\]

Therefore, the length of the red line segment is \(\sqrt{40}\) or approximately 6.32 units.

Now, let's evaluate the statements provided:

1. **The length is 40 units: find the sum of x and y coordinates and square root the answer.**
- This statement is **false**. We did not find the length using this sum method, and clearly \(d = \sqrt{40} \neq 40\).

2. **The length is \(\sqrt{260}\): find the difference of the x coordinates and the difference of the y coordinates and subtract those differences.**
- This statement is **false**. We calculated the distance as \(\sqrt{40}\), not \(\sqrt{260}\).

3. **The length is \(\sqrt{40}\) units:**
- This statement is **true**. As calculated, the length is indeed \(\sqrt{40}\).

4. **The length is 130:**
- This statement is **false**. The computed length does not match this value.

In conclusion, the only true statement is the third: **The length is \(\sqrt{40}\) units.**