Given the points A(0,0),B(3,1),and C(1,4), what is the measure of angle ABC?
the easiest way here would be to use vectors
u = AB = (3,1)
v = AC = (1,4)
u•v = |u| |v| cosθ
so,
cosθ = (3+4) / (√10 √17) = 7/√170 = 0.5368
θ = 57.5°
Or,
AB slope is 1/3
AC slope is 4
tanθ = (4 - 1/3) / (1+4/3) = 11/7 = 1.5714
θ = 57.5°
Oops. That's angle BAC.
But that's the method; adjust the numbers
To find the measure of angle ABC, we can use the concept of slope.
First, let's find the slopes of line AB and line BC. The slope of a line passing through points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
For line AB (points A(0,0) and B(3,1)):
slope_AB = (1 - 0) / (3 - 0) = 1/3
For line BC (points B(3,1) and C(1,4)):
slope_BC = (4 - 1) / (1 - 3) = 3/-2 = -3/2
Next, let's find the angles created by these slopes.
The measure of an angle formed by two lines with slopes m1 and m2 can be determined by using the formula:
angle = arctan |(m2 - m1) / (1 + m1 * m2)|
Using this formula, we can find the measure of angle ABC:
angle_ABC = arctan |((-3/2) - (1/3)) / (1 + (1/3) * (-3/2))|
Now, let's plug in the values and calculate:
angle_ABC = arctan |((-3/2) - (1/3)) / (1 + (1/3) * (-3/2))|
angle_ABC = arctan |(-9/6 - 2/6) / (6/6 - 9/6)|
angle_ABC = arctan |-11/6 / (-3/6)|
angle_ABC = arctan |11/3|
To find the measure of the angle in degrees, we can use the inverse tangent function on a calculator or online tool:
angle_ABC ≈ 77.95 degrees
So, the measure of angle ABC is approximately 77.95 degrees.