Find the distance between the pairs of points :(-5, 7); (-1, 3)

Q: Find the distance between the pairs of points :(-5, 7); (-1, 3)

To find the distance between the points $(-5, 7)$ and $(-1, 3)$, we use the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here, $(x_1, y_1) = (-5, 7)$ and $(x_2, y_2) = (-1, 3)$. Substituting these values into the formula, we have: \[ d = \sqrt{((-1) - (-5))^2 + (3 - 7)^2} \] Simplifying inside the parentheses: \[ d = \sqrt{(-1 + 5)^2 + (3 - 7)^2} \] \[ d = \sqrt{(4)^2 + (-4)^2} \] \[ d = \sqrt{16 + 16} \] \[ d = \sqrt{32} \] \[ d = \sqrt{16 \times 2} \] \[ d = \sqrt{16} \times \sqrt{2} \] \[ d = 4\sqrt{2} \] Thus, the distance between the points is $4\sqrt{2}$, which corresponds to Option 3. **Explanation of Incorrect Options:** 1. **Option 1: $2\sqrt{2}$** - This option is incorrect because the calculation of the distance results in $4\sqrt{2}$, not $2\sqrt{2}$. The error here would be in underestimating the distance by a factor of 2. 2. **Option 2: $-2\sqrt{2}$** - This option is incorrect because distances cannot be negative. The distance formula always yields a non-negative result, and the correct calculation gives a positive $4\sqrt{2}$. 3. **Option 4: $-4\sqrt{2}$** - Similar to Option 2, this option is incorrect because distances cannot be negative. The correct distance is $4\sqrt{2}$, which is positive. In conclusion, Option 3 is the correct answer because it accurately reflects the calculated distance between the given points using the distance formula.

Question

Accepted Answer

To find the distance between the points $(-5, 7)$ and $(-1, 3)$, we use the distance formula, which is given by:

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

Here, $(x_1, y_1) = (-5, 7)$ and $(x_2, y_2) = (-1, 3)$.

Substituting these values into the formula, we have:

$$ 
d = \sqrt{((-1) - (-5))^2 + (3 - 7)^2} 
$$

Simplifying inside the parentheses:

$$ 
d = \sqrt{(-1 + 5)^2 + (3 - 7)^2} 
$$

$$ 
d = \sqrt{(4)^2 + (-4)^2} 
$$

$$ 
d = \sqrt{16 + 16} 
$$

$$ 
d = \sqrt{32} 
$$

$$ 
d = \sqrt{16 \times 2} 
$$

$$ 
d = \sqrt{16} \times \sqrt{2} 
$$

$$ 
d = 4\sqrt{2} 
$$

Thus, the distance between the points is $4\sqrt{2}$, which corresponds to Option 3.

**Explanation of Incorrect Options:**

1. **Option 1: $2\sqrt{2}$** - This option is incorrect because the calculation of the distance results in $4\sqrt{2}$, not $2\sqrt{2}$. The error here would be in underestimating the distance by a factor of 2.

2. **Option 2: $-2\sqrt{2}$** - This option is incorrect because distances cannot be negative. The distance formula always yields a non-negative result, and the correct calculation gives a positive $4\sqrt{2}$.

3. **Option 4: $-4\sqrt{2}$** - Similar to Option 2, this option is incorrect because distances cannot be negative. The correct distance is $4\sqrt{2}$, which is positive.

In conclusion, Option 3 is the correct answer because it accurately reflects the calculated distance between the given points using the distance formula.