cosine law
Given the points A(0,0) B (3,1) and C (1,4) what is the measure of angle ABC
To find the measure of angle ABC, we can use the cosine law. The cosine law states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides multiplied by the cosine of the angle between them.
In this case, we can use the points A(0,0), B(3,1), and C(1,4) to find the lengths of the sides of the triangle.
First, let's find the length of side AB. Using the distance formula, we have:
AB = sqrt((3 - 0)^2 + (1 - 0)^2)
= sqrt(9 + 1)
= sqrt(10)
Next, let's find the length of side BC:
BC = sqrt((1 - 3)^2 + (4 - 1)^2)
= sqrt((-2)^2 + 3^2)
= sqrt(4 + 9)
= sqrt(13)
Finally, let's find the length of side AC:
AC = sqrt((1 - 0)^2 + (4 - 0)^2)
= sqrt(1^2 + 4^2)
= sqrt(1 + 16)
= sqrt(17)
Now that we have the lengths of the sides AB, BC, and AC, we can use the cosine law to find the measure of angle ABC. The cosine law formula is:
AB^2 = AC^2 + BC^2 - 2(AC)(BC)cos(ABC)
Substituting the values we have:
(√10)^2 = (√17)^2 + (√13)^2 - 2(√17)(√13)cos(ABC)
10 = 17 + 13 - 2√221cos(ABC)
10 = 30 - 2√221cos(ABC)
2√221cos(ABC) = 20
cos(ABC) = 20 / (2√221)
= 10 / √221
Therefore, the measure of angle ABC is equal to the inverse cosine (arccos) of 10 / √221.