Answer :
To calculate the distance between the points [tex]\( Q = (-1, -1) \)[/tex] and [tex]\( C = (6, -7) \)[/tex] in the coordinate plane, we can use the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's apply this step-by-step:
1. Identify the coordinates of the points:
- Point [tex]\( Q \)[/tex] has coordinates [tex]\((-1, -1)\)[/tex].
- Point [tex]\( C \)[/tex] has coordinates [tex]\((6, -7)\)[/tex].
2. Calculate the differences in the x-coordinates and y-coordinates:
- Difference in the x-coordinates: [tex]\( x_2 - x_1 = 6 - (-1) = 6 + 1 = 7 \)[/tex]
- Difference in the y-coordinates: [tex]\( y_2 - y_1 = -7 - (-1) = -7 + 1 = -6 \)[/tex]
3. Square the differences:
- Square of the difference in the x-coordinates: [tex]\( 7^2 = 49 \)[/tex]
- Square of the difference in the y-coordinates: [tex]\( (-6)^2 = 36 \)[/tex]
4. Sum the squares of the differences:
- Sum of the squares: [tex]\( 49 + 36 = 85 \)[/tex]
5. Take the square root of the sum to find the distance:
- Distance: [tex]\( \sqrt{85} \)[/tex]
So, the distance between the points [tex]\( Q \)[/tex] and [tex]\( C \)[/tex] is [tex]\( \sqrt{85} \)[/tex].
Thus, the exact answer is:
[tex]\[ \text{Distance} = \sqrt{85} \][/tex]
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's apply this step-by-step:
1. Identify the coordinates of the points:
- Point [tex]\( Q \)[/tex] has coordinates [tex]\((-1, -1)\)[/tex].
- Point [tex]\( C \)[/tex] has coordinates [tex]\((6, -7)\)[/tex].
2. Calculate the differences in the x-coordinates and y-coordinates:
- Difference in the x-coordinates: [tex]\( x_2 - x_1 = 6 - (-1) = 6 + 1 = 7 \)[/tex]
- Difference in the y-coordinates: [tex]\( y_2 - y_1 = -7 - (-1) = -7 + 1 = -6 \)[/tex]
3. Square the differences:
- Square of the difference in the x-coordinates: [tex]\( 7^2 = 49 \)[/tex]
- Square of the difference in the y-coordinates: [tex]\( (-6)^2 = 36 \)[/tex]
4. Sum the squares of the differences:
- Sum of the squares: [tex]\( 49 + 36 = 85 \)[/tex]
5. Take the square root of the sum to find the distance:
- Distance: [tex]\( \sqrt{85} \)[/tex]
So, the distance between the points [tex]\( Q \)[/tex] and [tex]\( C \)[/tex] is [tex]\( \sqrt{85} \)[/tex].
Thus, the exact answer is:
[tex]\[ \text{Distance} = \sqrt{85} \][/tex]
