Determine the diameter of the largest circle that can be drawn between the parallel lines y=2x and y=2x-5.
What is the distance between them?
find the perpendicular line
y=-1/2 x + b
pick a point on the first line: say, 1,2
2= -1/2 *1+b
b=2.5
y= -.5x+2.5 is the perpendicular line.
now, what point does this line intersect the second parallel line?
y=-.5x+2.5
y=2x-5
or 2x-5=-.5x+2.5
2.5x=7.5
x=3
y=2x-5=1
distance between points 1,2 and 3,1
d= sqrt (2^2 + 1^2)=sqrt5
check all that math, I did it in my head.
To determine the diameter of the largest circle that can be drawn between the parallel lines y = 2x and y = 2x - 5, we need to find the distance between the two lines.
Let's first find the equations of the lines in the general form (Ax + By + C = 0):
Line 1: y = 2x
Rewriting the equation, we have 2x - y = 0.
So, A = 2, B = -1, and C = 0.
Line 2: y = 2x - 5
Rewriting the equation, we have 2x - y + 5 = 0.
So, A = 2, B = -1, and C = 5.
The distance between two parallel lines (Ax + By + C₁ = 0) and (Ax + By + C₂ = 0) is given by the formula:
Distance = |C₂ - C₁| / √(A² + B²)
In this case, the distance between the two lines is:
Distance = |5 - 0| / √(2² + (-1)²)
= 5 / √(4 + 1)
= 5 / √5
Simplifying, we get:
Distance = 5 / √5
= 5 / √5 * (√5 / √5)
= 5√5 / 5
= √5
Since the diameter of a circle is equal to twice its radius, the diameter of the largest circle that can be drawn between the parallel lines y = 2x and y = 2x - 5 is equal to 2 times the distance between the lines:
Diameter = 2 * √5
Diameter ≈ 4.47