Questions: Question 4 (Mandatory) (4 points) For the points P and Q, find the distance d(P, Q). P(6,-7), Q(2,-5) 6 12 2 sqrt(5) 12 sqrt(3)

Transcript text: Question 4 (Mandatory) (4 points) For the points $P$ and $Q$, find the distance $d(P, Q)$. \[ P(6,-7), Q(2,-5) \] 6 12 $2 \sqrt{5}$ $12 \sqrt{3}$

Solution

Solution Steps

Step 1: Identify the coordinates of points $ P $ and $ Q $

The coordinates of point $ P $ are $ (6, -7) $, and the coordinates of point $ Q $ are $ (2, -5) $.

Step 2: Apply the distance formula

The distance $ d(P, Q) $ between two points $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ is given by: \[ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Substitute the coordinates into the distance formula

Substitute $ x_1 = 6 $, $ y_1 = -7 $, $ x_2 = 2 $, and $ y_2 = -5 $ into the formula: \[ d(P, Q) = \sqrt{(2 - 6)^2 + (-5 - (-7))^2} \]

Step 4: Simplify the expressions inside the square root

Calculate the differences: \[ 2 - 6 = -4 \quad \text{and} \quad -5 - (-7) = 2 \] Now, square these differences: \[ (-4)^2 = 16 \quad \text{and} \quad 2^2 = 4 \]

Step 5: Add the squared differences and take the square root

Add the squared differences: \[ 16 + 4 = 20 \] Take the square root of 20: \[ \sqrt{20} = 2\sqrt{5} \]