Given the points A(0,0)B(3,1), and C(1,4), what is the measure of angle ABC?
I've done this already, but maybe you didn't see it or didn't like it. What topic are you studying? There are various ways to go about solving it, and I'd hate to use a method that will not be acceptable as part of the class.
To find the measure of angle ABC, you will need to use the concept of vectors and dot product. The dot product of two vectors can be used to find the angle between them.
First, we need to find the vectors AB and BC using their coordinates:
Vector AB = B - A = (3, 1) - (0, 0) = (3, 1)
Vector BC = C - B = (1, 4) - (3, 1) = (-2, 3)
Next, we'll calculate the dot product of AB and BC using the formula:
Dot product (AB · BC) = (x₁ * x₂) + (y₁ * y₂)
AB · BC = (3 * -2) + (1 * 3) = -6 + 3 = -3
Now, let's find the magnitudes (lengths) of AB and BC using the formula:
Magnitude (|v|) = √(x² + y²)
|AB| = √(3² + 1²) = √(9 + 1) = √10
|BC| = √((-2)² + 3²) = √(4 + 9) = √13
Finally, we can find the angle between the two vectors using the dot product and magnitudes:
Angle ABC = cos⁻¹((AB · BC) / (|AB| * |BC|))
= cos⁻¹((-3) / (√10 * √13))
By plugging the values into a calculator, you will get the answer in radians. To convert it into degrees, multiply by (180/π).