Question
Find the distance between these two points using the Pythagorean theorem. (-3,5) and (7,-1) Which choice is correct?
square root 3^2 - 5^2= square root of 20
square root 6^2 + 3^2= square root 25
5^2+7^2= square root 136
square root (6^2+10^2)= square root 136
square root 3^2 - 5^2= square root of 20
square root 6^2 + 3^2= square root 25
5^2+7^2= square root 136
square root (6^2+10^2)= square root 136
Answers
Answered by
GPT-4o mini
To find the distance between the two points \((-3, 5)\) and \((7, -1)\) using the Pythagorean theorem, we first need to determine the differences in the x-coordinates and y-coordinates:
1. Calculate the difference in the x-coordinates:
\[
x_2 - x_1 = 7 - (-3) = 7 + 3 = 10
\]
2. Calculate the difference in the y-coordinates:
\[
y_2 - y_1 = -1 - 5 = -6
\]
Now we have the differences as \(10\) (for the x-coordinates) and \(-6\) (for the y-coordinates).
Next, we can use the Pythagorean theorem to find the distance \(d\):
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{10^2 + (-6)^2} = \sqrt{100 + 36} = \sqrt{136}
\]
Now, looking at the choices provided:
- **square root 3^2 - 5^2 = square root of 20** (Incorrect)
- **square root 6^2 + 3^2 = square root 25** (Incorrect)
- **5^2 + 7^2 = square root 136** (Incorrect operation, should not equal square root 136)
- **square root (6^2 + 10^2) = square root 136** (Incorrect operation for distance but correctly leads to \( \sqrt{136} \))
It looks like we reached \( \sqrt{136} \) using our calculations, but none of the options perfectly match our derivation. However, the closest and somewhat related answer you provided in terms of match to what we derived is:
**square root (6^2 + 10^2) = square root 136** (with corrections regarding the meaning behind it, primarily since it's about distance, this is the best match).
So, if this is the one that you've intended to mean in a way corresponding closely to the correct operation derived, that would be the best fit among the options.
1. Calculate the difference in the x-coordinates:
\[
x_2 - x_1 = 7 - (-3) = 7 + 3 = 10
\]
2. Calculate the difference in the y-coordinates:
\[
y_2 - y_1 = -1 - 5 = -6
\]
Now we have the differences as \(10\) (for the x-coordinates) and \(-6\) (for the y-coordinates).
Next, we can use the Pythagorean theorem to find the distance \(d\):
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{10^2 + (-6)^2} = \sqrt{100 + 36} = \sqrt{136}
\]
Now, looking at the choices provided:
- **square root 3^2 - 5^2 = square root of 20** (Incorrect)
- **square root 6^2 + 3^2 = square root 25** (Incorrect)
- **5^2 + 7^2 = square root 136** (Incorrect operation, should not equal square root 136)
- **square root (6^2 + 10^2) = square root 136** (Incorrect operation for distance but correctly leads to \( \sqrt{136} \))
It looks like we reached \( \sqrt{136} \) using our calculations, but none of the options perfectly match our derivation. However, the closest and somewhat related answer you provided in terms of match to what we derived is:
**square root (6^2 + 10^2) = square root 136** (with corrections regarding the meaning behind it, primarily since it's about distance, this is the best match).
So, if this is the one that you've intended to mean in a way corresponding closely to the correct operation derived, that would be the best fit among the options.