To find the distance between the points (-3, 5) and (7, -1), we can use the distance formula, which is derived from the Pythagorean Theorem. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, the coordinates are:
- \( (x_1, y_1) = (-3, 5) \)
- \( (x_2, y_2) = (7, -1) \)
Now, calculate the differences:
\[ x_2 - x_1 = 7 - (-3) = 7 + 3 = 10 \]
\[ y_2 - y_1 = -1 - 5 = -6 \]
Then, plug these values into the distance formula:
\[ d = \sqrt{(10)^2 + (-6)^2} \]
Calculating the squares:
\[ (10)^2 = 100 \] \[ (-6)^2 = 36 \]
Now add them together:
\[ d = \sqrt{100 + 36} = \sqrt{136} \]
So, the distance between the two points is \( \sqrt{136} \).
Now let's examine the response options to find the correct choice:
-
(62+102)−−−−−−−−−√=136−−−√
- This option states \( \sqrt{6^2 + 10^2} = \sqrt{136} \), which is not correct since we found the differences to be 10 and -6, not 6 and 10.
-
32−−√−52=20−−√
- This option doesn't make sense in the context of distance calculation. It does not correlate to our computation.
-
52+72=136−−−√
- This option states \( 5^2 + 7^2 = \sqrt{136} \) which is also incorrect because we have \( 10 \) as first and \(-6\) as second value.
-
62−−√+32=25−−√
- This option does not make sense in the context of distance.
None of the response options correctly represent the distance calculation we performed using the points (-3, 5) and (7, -1).
The correct answer should have been \( \sqrt{(10)^2 + (-6)^2} = \sqrt{136} \). If the options contain an error, it's advisable to check again or provide a correct set of options.