Given the points A(0,0), B(4, 3) and C(2,9), what is the measure of angle ABC?
Using Law of Cosines.
AB = 5
BC = √40
AC = √85, so
85 = 25+40-2(5)(√40)cos(ABC)
20 = -20√10 cos(ABC
cos(ABC) = -1/√10
ABC = 108.43°
To find the measure of angle ABC using the Law of Cosines, we can start by calculating the distances between the three points A, B, and C.
1. Distance between A and B:
Using the distance formula, we can calculate the distance between points A(0,0) and B(4,3) as follows:
AB = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((4 - 0)² + (3 - 0)²)
= √(4² + 3²)
= √(16 + 9)
= √25
= 5
2. Distance between B and C:
Similarly, using the distance formula, we can calculate the distance between points B(4,3) and C(2,9) as follows:
BC = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((2 - 4)² + (9 - 3)²)
= √((-2)² + 6²)
= √(4 + 36)
= √40
= 2√10
3. Distance between C and A:
Using the distance formula, we can calculate the distance between points C(2,9) and A(0,0) as follows:
CA = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((0 - 2)² + (0 - 9)²)
= √((-2)² + (-9)²)
= √(4 + 81)
= √85
Now that we know the lengths of the sides of triangle ABC, we can apply the Law of Cosines to find the measure of angle ABC.
The Law of Cosines states:
c² = a² + b² - 2ab * cosine(C)
In our case, c refers to the side opposite angle ABC, a refers to the side opposite angle BAC, b refers to the side opposite angle BCA, and C refers to the measure of angle ABC.
Plugging in the values we calculated:
(2√10)² = 5² + (√85)² - 2 * 5 * √85 * cosine(C)
Simplifying the equation:
40 = 25 + 85 - 10√85 * cosine(C)
40 = 110 - 10√85 * cosine(C)
Now we can isolate cosine(C):
10√85 * cosine(C) = 110 - 40
10√85 * cosine(C) = 70
Solving for cosine(C):
cosine(C) = 70 / (10√85)
cosine(C) = 7 / (√85)
To find the measure of angle ABC, we can take the inverse cosine (arccosine) of cosine(C):
angle ABC = arccos(7 / √85)
By evaluating this expression using a calculator or trigonometric table, we can find the measure of angle ABC in degrees.