The distance between the points (-4, 7) and (2, -5) is :

Q: The distance between the points (-4, 7) and (2, -5) is :

To determine the distance between the points $(-4, 7)$ and $ (2, -5) $, we use the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the given points $(-4, 7)$ and $ (2, -5) $ into the formula, we have: \[ x_1 = -4, \quad y_1 = 7, \quad x_2 = 2, \quad y_2 = -5 \] \[ d = \sqrt{(2 - (-4))^2 + (-5 - 7)^2} \] \[ d = \sqrt{(2 + 4)^2 + (-5 - 7)^2} \] \[ d = \sqrt{6^2 + (-12)^2} \] \[ d = \sqrt{36 + 144} \] \[ d = \sqrt{180} \] \[ d = \sqrt{36 \times 5} \] \[ d = \sqrt{36} \times \sqrt{5} \] \[ d = 6\sqrt{5} \] Thus, the distance between the points is $6\sqrt{5} \text{ units}$, which corresponds to Option 4. ### Explanation of Incorrect Options: 1. **Option 1: $5\sqrt{6} \text{ units}$** This option is incorrect because the calculation of the distance does not result in $5\sqrt{6}$. The correct simplification of $\sqrt{180}$ is $6\sqrt{5}$, not $5\sqrt{6}$. 2. **Option 2: $\sqrt{56} \text{ units}$** This option is incorrect because $\sqrt{56}$ does not match the calculated distance. The expression $\sqrt{180}$ simplifies to $6\sqrt{5}$, not $\sqrt{56}$. 3. **Option 3: $4\sqrt{5} \text{ units}$** This option is incorrect because the simplification of $\sqrt{180}$ results in $6\sqrt{5}$, not $4\sqrt{5}$. The factorization and simplification process clearly leads to $6\sqrt{5}$. In conclusion, the correct answer is Option 4: $6\sqrt{5} \text{ units}$, as it accurately represents the distance between the given points according to the distance formula.

Question

Accepted Answer

To determine the distance between the points $(-4, 7)$ and $ (2, -5) $, we use the distance formula, which is given by:

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

Substituting the given points $(-4, 7)$ and $ (2, -5) $ into the formula, we have:

$$
x_1 = -4, \quad y_1 = 7, \quad x_2 = 2, \quad y_2 = -5
$$

$$
d = \sqrt{(2 - (-4))^2 + (-5 - 7)^2}
$$

$$
d = \sqrt{(2 + 4)^2 + (-5 - 7)^2}
$$

$$
d = \sqrt{6^2 + (-12)^2}
$$

$$
d = \sqrt{36 + 144}
$$

$$
d = \sqrt{180}
$$

$$
d = \sqrt{36 \times 5}
$$

$$
d = \sqrt{36} \times \sqrt{5}
$$

$$
d = 6\sqrt{5}
$$

Thus, the distance between the points is $6\sqrt{5} \text{ units}$, which corresponds to Option 4.

### Explanation of Incorrect Options:

1. **Option 1: $5\sqrt{6} \text{ units}$**

This option is incorrect because the calculation of the distance does not result in $5\sqrt{6}$. The correct simplification of $\sqrt{180}$ is $6\sqrt{5}$, not $5\sqrt{6}$.

2. **Option 2: $\sqrt{56} \text{ units}$**

This option is incorrect because $\sqrt{56}$ does not match the calculated distance. The expression $\sqrt{180}$ simplifies to $6\sqrt{5}$, not $\sqrt{56}$.

3. **Option 3: $4\sqrt{5} \text{ units}$**

This option is incorrect because the simplification of $\sqrt{180}$ results in $6\sqrt{5}$, not $4\sqrt{5}$. The factorization and simplification process clearly leads to $6\sqrt{5}$.

In conclusion, the correct answer is Option 4: $6\sqrt{5} \text{ units}$, as it accurately represents the distance between the given points according to the distance formula.