Use vectors to find the interior angles of the triangle given the following sets of vertices. Round your answer, in degrees, \[ (7,0),(-7,5),(-1,-8) \]
Solution
Step 1 ：Given the vertices of the triangle A = (7, 0), B = (-7, 5), and C = (-1, -8).
Step 2 ：Calculate the vectors AB, BC, and CA. AB = B - A = (-14, 5), BC = C - B = (6, -13), and CA = A - C = (8, 8).
Step 3 ：Calculate the magnitudes of these vectors. The magnitude of AB is approximately 14.87, the magnitude of BC is approximately 14.32, and the magnitude of CA is approximately 11.31.
Step 4 ：Calculate the dot products of the vectors. The dot product of AB and BC is -149, the dot product of BC and CA is -56, and the dot product of CA and AB is -72.
Step 5 ：Use the dot product formula to find the angles. The angle between AB and BC is approximately \(134^\circ\), the angle between BC and CA is approximately \(110^\circ\), and the angle between CA and AB is approximately \(115^\circ\).
Step 6 ：Final Answer: The interior angles of the triangle are approximately \(\boxed{134^\circ}\), \(\boxed{110^\circ}\), and \(\boxed{115^\circ}\).
