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)

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 $2 \sqrt{5}$ $12 \sqrt{3}$
failed

Solution

failed
failed

Solution Steps

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

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

Step 2: Apply the distance formula

The distance d(P,Q) d(P, Q) between two points P(x1,y1) P(x_1, y_1) and Q(x2,y2) Q(x_2, y_2) is given by: d(P,Q)=(x2x1)2+(y2y1)2 d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Step 3: Substitute the coordinates into the distance formula

Substitute x1=6 x_1 = 6 , y1=7 y_1 = -7 , x2=2 x_2 = 2 , and y2=5 y_2 = -5 into the formula: d(P,Q)=(26)2+(5(7))2 d(P, Q) = \sqrt{(2 - 6)^2 + (-5 - (-7))^2}

Step 4: Simplify the expressions inside the square root

Calculate the differences: 26=4and5(7)=2 2 - 6 = -4 \quad \text{and} \quad -5 - (-7) = 2 Now, square these differences: (4)2=16and22=4 (-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 16 + 4 = 20 Take the square root of 20: 20=25 \sqrt{20} = 2\sqrt{5}

Final Answer

The distance d(P,Q) d(P, Q) is 25 \boxed{2\sqrt{5}} .

Was this solution helpful?
failed
Unhelpful
failed
Helpful