Given the points (A) 0,0 (B) 4,3 (C) 2,9 what is the measure of angle ABC
AB = 5
AC = √85
BC = √40
Call the angle x. Then by the law of cosines,
40 = 25+85-2*5√85 cos(x)
To find the measure of angle ABC, we can use trigonometry. Here are the steps:
Step 1: Find the lengths of sides AB, BC, and AC.
- AB = √[(x2 - x1)^2 + (y2 - y1)^2]
- AB = √[(4 - 0)^2 + (3 - 0)^2]
- AB = √[16 + 9]
- AB = √25
- AB = 5
- BC = √[(x2 - x1)^2 + (y2 - y1)^2]
- BC = √[(2 - 4)^2 + (9 - 3)^2]
- BC = √[4 + 36]
- BC = √40
- BC = 2√10
- AC = √[(x2 - x1)^2 + (y2 - y1)^2]
- AC = √[(2 - 0)^2 + (9 - 0)^2]
- AC = √[4 + 81]
- AC = √85
Step 2: Apply the Law of Cosines to find angle ABC.
- cos(ABC) = [(AB^2 + BC^2 - AC^2) / (2 * AB * BC)]
- cos(ABC) = [(5^2 + (2√10)^2 - (√85)^2) / (2 * 5 * 2√10)]
- cos(ABC) = [(25 + 40 - 85) / (10√10)]
- cos(ABC) = [-20 / (10√10)]
- cos(ABC) = -2 / √10
Step 3: Find the measure of angle ABC.
- ABC = arccos(cos(ABC))
- ABC = arccos(-2 / √10)
Using a calculator to evaluate arccos(-2 / √10), we find that the measure of angle ABC is approximately 128.66 degrees.
To find the measure of angle ABC, we can use the concept of trigonometry.
First, let's find the lengths of sides AB, BC, and AC using the distance formula:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(4 - 0)^2 + (3 - 0)^2]
= √[16 + 9]
= √25
= 5
BC = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - 4)^2 + (9 - 3)^2]
= √[(-2)^2 + 6^2]
= √[4 + 36]
= √40
= 2√10
AC = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - 0)^2 + (9 - 0)^2]
= √[2^2 + 9^2]
= √[4 + 81]
= √85
Now, to find the measure of angle ABC, we can use the law of cosines:
cos(∠ABC) = (AB^2 + BC^2 - AC^2) / (2 * AB * BC)
cos(∠ABC) = (5^2 + (2√10)^2 - √85^2) / (2 * 5 * 2√10)
cos(∠ABC) = (25 + 40 - 85) / (10√10)
cos(∠ABC) = (65 - 85) / (10√10)
cos(∠ABC) = -20 / (10√10)
Now, let's calculate the value of cos(∠ABC):
cos(∠ABC) ≈ -0.632
To find the measure of ∠ABC, we can use the inverse cosine function (cos^(-1)):
∠ABC ≈ cos^(-1)(-0.632)
Using a calculator, we find:
∠ABC ≈ 133.61 degrees
Therefore, the measure of angle ABC is approximately 133.61 degrees.