Find the interior angles of the triangle with vertices (-4,1), (1,-6), and (5,-1)
If we label the points A,B,C, then
AB=√84
BC=√41
AC=√85
85 = 84+41-2√84√41 cosB
Now you can use law of sine or cosine for one other angle, and then the angles sum to 180.
To find the interior angles of a triangle, we can use the formula for the angle between two vectors. Here's how you can do that:
1. Start by finding the vectors formed by the sides of the triangle. Let's call the vertices A(-4, 1), B(1, -6), and C(5, -1).
2. Find the vectors AB, AC, and BC by subtracting the coordinates of the endpoints. For example, vector AB can be found by subtracting the coordinates of point A from the coordinates of point B: AB = B - A.
AB = (1, -6) - (-4, 1) = (1 + 4, -6 - 1) = (5, -7)
AC = (5, -1) - (-4, 1) = (5 + 4, -1 - 1) = (9, -2)
BC = (5, -1) - (1, -6) = (5 - 1, -1 + 6) = (4, 5)
3. Use the dot product formula to find the angle between two vectors. The formula for the dot product of two vectors A and B is given by:
A · B = |A| * |B| * cos(angle)
We can rearrange the formula to solve for the angle:
angle = cos⁻¹((A · B) / (|A| * |B|))
4. Calculate the dot product of AB and AC. Then calculate the dot product of AC and BC. Finally, calculate the dot product of AB and BC.
AB · AC = (5 * 9) + (-7 * -2) = 45 + 14 = 59
AC · BC = (9 * 4) + (-2 * 5) = 36 - 10 = 26
AB · BC = (5 * 4) + (-7 * 5) = 20 - 35 = -15
5. Calculate the magnitudes (lengths) of vectors AB, AC, and BC using the formula |A| = √(A₁² + A₂²).
|AB| = √(5² + (-7)²) = √(25 + 49) = √74
|AC| = √(9² + (-2)²) = √(81 + 4) = √85
|BC| = √(4² + 5²) = √(16 + 25) = √41
6. Substitute the calculated values into the angle formula for each pair of vectors to find the angles.
Angle A = cos⁻¹((AB · AC) / (|AB| * |AC|))
Angle B = cos⁻¹((AC · BC) / (|AC| * |BC|))
Angle C = cos⁻¹((AB · BC) / (|AB| * |BC|))
Angle A = cos⁻¹(59 / (√74 * √85))
Angle B = cos⁻¹(26 / (√85 * √41))
Angle C = cos⁻¹(-15 / (√74 * √41))
7. Use a calculator (or a programming language) to find the arccosine (cos⁻¹) of the fraction. Make sure your calculator is set to degrees instead of radians to get the angle in degrees.
Angle A ≈ 87.63°
Angle B ≈ 51.59°
Angle C ≈ 40.78°
Therefore, the interior angles of the triangle with vertices (-4, 1), (1, -6), and (5, -1) are approximately:
Angle A ≈ 87.63°
Angle B ≈ 51.59°
Angle C ≈ 40.78°