To determine the size of the acute angle between the lines AO and AB, where A is the point (-2, 3) and B is the point (3, -2), we can use the tangent rule, which states that the tangent of the angle between two lines is equal to the absolute value of the slope difference of the two lines divided by 1 plus the product of their slopes.
The slope of the line AO, denoted as m_AO, is the slope of the line passing through points A and the origin (O), which is (0, 0). The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
For line AO:
m_AO = (0 - 3) / (0 - (-2)) = -3 / 2
Now, let's find the slope of line AB:
m_AB = (-2 - 3) / (3 - (-2)) = -5 / 5 = -1
Now, we can use the tangent rule:
tan(θ) = |(m_AO - m_AB) / (1 + m_AO * m_AB)|
tan(θ) = |((-3/2) - (-1)) / (1 + (-3/2) * (-1))|
tan(θ) = |((-3/2) + 1) / (1 + (3/2))|
tan(θ) = |(-3/2 + 2/2) / (2/2 + 3/2)|
tan(θ) = |(-1/2) / (5/2)|
Now, calculate the absolute value of this expression:
tan(θ) = |-1/2| / |5/2|
tan(θ) = (1/2) / (5/2)
Now, simplify by multiplying both the numerator and denominator by 2:
tan(θ) = (1/2) * (2/5)
tan(θ) = 1/5
Now, to find the acute angle θ, take the arctan (inverse tangent) of 1/5:
θ = arctan(1/5)
Using a calculator or a trigonometric table, find the value of arctan(1/5). This will give you the size of the acute angle between the lines AO and AB. The result will be in degrees.