Question

use the distance formula. find the distance between the following pairs of points:

1. a(-4,0) and b(2,0)
2. p(3,5) and q(3,1)
3. s(-4,5) and t(6,-5)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate distance for points A(-4,0) and B(2,0)

Here $x_1=-4,y_1 = 0,x_2=2,y_2 = 0$. Substitute into the formula:
$d_{AB}=\sqrt{(2-(-4))^2+(0 - 0)^2}=\sqrt{(2 + 4)^2}=\sqrt{6^2}=6$.

Step3: Calculate distance for points P(3,5) and Q(3,1)

Here $x_1 = 3,y_1=5,x_2 = 3,y_2=1$. Substitute into the formula:
$d_{PQ}=\sqrt{(3 - 3)^2+(1 - 5)^2}=\sqrt{0+( - 4)^2}=\sqrt{16}=4$.

Step4: Calculate distance for points S(-4,5) and T(6,-5)

Here $x_1=-4,y_1 = 5,x_2=6,y_2=-5$. Substitute into the formula:
$d_{ST}=\sqrt{(6-(-4))^2+((-5)-5)^2}=\sqrt{(6 + 4)^2+(-10)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}$.

Answer:

1. Distance between A(-4,0) and B(2,0) is 6.
2. Distance between P(3,5) and Q(3,1) is 4.
3. Distance between S(-4,5) and T(6,-5) is $10\sqrt{2}$.