Question

for the points p and q, find the distance ( d(p,q) ).
( p(3, -2) ), ( q(7, -5) )

a. ( sqrt{7} )
b. ( 7 )
c. ( 25 )
d. ( 5 )

Explanation:

Step1: Recall the distance formula

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

Step2: Identify the coordinates

For \( P(3, - 2) \) and \( Q(7, - 5) \), we have \( x_1 = 3,y_1=-2,x_2 = 7,y_2=-5 \).

Step3: Substitute into the formula

First, calculate \( x_2 - x_1=7 - 3 = 4 \) and \( y_2 - y_1=-5-(-2)=-5 + 2=-3 \).
Then, \( (x_2 - x_1)^2=4^2 = 16 \) and \( (y_2 - y_1)^2=(-3)^2 = 9 \).
Next, \( (x_2 - x_1)^2+(y_2 - y_1)^2=16 + 9=25 \).
Finally, \( d(P, Q)=\sqrt{25}=5 \).

Answer:

D. 5