find the distance AB between points A and B (5,4) when lines y=x+1 and 3x=4y=11 intersect at point A
First, we need to find the coordinates of the point where the two lines intersect (point A).
Given:
y = x + 1
3x = 4y = 11
Substitute y = x + 1 into the second equation:
3x = 4(x + 1) = 11
3x = 4x + 4
-x = 4
x = -4
Now, plug x = -4 back into y = x + 1:
y = -4 + 1
y = -3
Therefore, point A is located at coordinates (-4, -3).
Now, we can calculate the distance between point A (-4, -3) and point B (5, 4) using the distance formula:
AB = sqrt((x2-x1)^2 + (y2-y1)^2)
AB = sqrt((5-(-4))^2 + (4-(-3))^2)
AB = sqrt((5+4)^2 + (4+3)^2)
AB = sqrt(9^2 + 7^2)
AB = sqrt(81 + 49)
AB = sqrt(130) ≈ 11.4
Therefore, the distance AB between points A and B is approximately 11.4 units.