We have to determine the angle between the lines y - 12 = 0 and 2x - 2y + 98 = 0.
We see that y -12 = 0 is a horizontal line. It forms an angle of 90 degrees with the x axis.
The slope of a line is the tangent of the angle it forms with the positive x-axis.
2x - 2y + 98 = 0
=> y = x + 49
The slope here is 1.
This implies that the line forms an angle arc tan 1 with the positive x-axis.
arc tan 1 = 45 degrees
Using the angles formed by the lines with the x-axis we get the angle between them as 90 - 45 = 45 degrees.
The acute angle between the given lines is 45 degrees.
The angle between two lines could be found using the formula:
tan x = (m2 -m1)/(1 + m1*m2)
m1 is the slope of the first line: y - 12 = 0 => y = 12 => m1 = 0
m2 is the slope of the 2nd line: 2x- 2y + 98 = 0
2y = 2x - 98
y = x - 49
m2 = 1
tan x = (1 - 0)/(1 + 1*0)
tan x = 1/1
tan x = 1
x = 45 degrees
The angle between the given lines measures 45 degrees.