One line passes through the points (1,9) and (2,6), and another line passes through (3,3) and (-1,5). Determine the acute angle between the two lines.
To determine the acute angle between two lines, we need to find the slopes of the two lines and use the formula for the angle between two lines.
Let's start by finding the slopes of the two lines. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
slope = (y2 - y1) / (x2 - x1)
Let's calculate the slopes of the two lines:
Line 1:
Using the points (1,9) and (2,6):
slope1 = (6 - 9) / (2 - 1) = -3
Line 2:
Using the points (3,3) and (-1,5):
slope2 = (5 - 3) / (-1 - 3) = 2 / -4 = -1/2
Now that we have the slopes of the two lines, we can use the formula to find the acute angle between them. The formula is:
angle = arctan((m2 - m1) / (1 + m1 * m2))
Let's substitute the slopes into the formula and calculate the acute angle:
angle = arctan((-1/2 - (-3)) / (1 + (-3) * (-1/2)))
Simplifying the expression:
angle = arctan((-1/2 + 3) / (1 + 3/2))
Further simplification:
angle = arctan((5/2) / (5/2))
angle = arctan(1)
Using a calculator, we find that arctan(1) ≈ 45 degrees.
Therefore, the acute angle between the two lines is approximately 45 degrees.