given the points A(0,0) ,B(4,3),C(2,9) what is the measure of < ABC?
let a=BC
b = AC
c = AB
find the length of each of the sides a, b, and c
the use the cosine law.
b^2 = a^2 + c^2 - 2ac cos B
solve for angle B
I will do one of the lengths,
a= √((2-4)^2 + 9-3)^2 ) = √(4+36) = √40
so would
b=the square root of 85 ?
To find the measure of angle ABC, we need to use the coordinates of points A, B, and C.
Step 1: Calculate the slopes of lines AB and BC.
The slope of line AB = (change in y-coordinate) / (change in x-coordinate) = (3 - 0) / (4 - 0) = 3/4.
The slope of line BC = (change in y-coordinate) / (change in x-coordinate) = (9 - 3) / (2 - 4) = 6/-2 = -3.
Step 2: Use the slopes to find the tangent of the angle ABC.
The tangent of angle ABC = (slope of AB - slope of BC) / (1 + slope of AB * slope of BC) = (3/4 - (-3)) / (1 + (3/4 * -3)) = (3/4 + 3) / (1 - (9/4)) = (15/4) / (-5/4) = -15/5 = -3.
Step 3: Find the angle measure using the inverse tangent function.
The measure of angle ABC = atan(tangent of angle ABC) = atan(-3) ≈ -1.249, or you can consider 180° - |result| ≈ 180° - |-1.249| = 180° - 1.249 ≈ 178.751.
So, the measure of angle ABC is approximately 178.751 degrees.
To find the measure of angle ABC, we can use the concept of slope.
Step 1: Find the slopes of the lines AB and BC.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1) / (x2 - x1).
Slope of AB:
(x1, y1) = (0, 0)
(x2, y2) = (4, 3)
Slope of AB = (3 - 0) / (4 - 0) = 3/4
Slope of BC:
(x1, y1) = (4, 3)
(x2, y2) = (2, 9)
Slope of BC = (9 - 3) / (2 - 4) = 6/-2 = -3
Step 2: Use the formula for the angle between two lines.
The formula to find the angle between two lines with slopes m1 and m2 is given by:
Angle = arctan(|(m2 - m1) / (1 + m1 * m2)|)
where arctan represents the inverse tangent function.
Angle = arctan(|(-3 - 3/4) / (1 + (-3/4) * (-3))|)
Angle = arctan(|(-12/4 - 3/4) / (1 + 9/4)|)
Angle = arctan(|(-15/4) / (13/4)|)
Angle = arctan(|-15/13|)
Angle ≈ 50.19 degrees
Therefore, the measure of angle ABC is approximately 50.19 degrees.