To find the value of C that makes angle ABC a right angle, we can use slope and midpoint formulas to find the equation of the perpendicular line passing through the midpoint of AB. By solving this equation, we find that the value of C is approximately 2.9333.
To find the value of C that makes angle ABC a right angle, we can use the properties of perpendicular lines. The slope of the line passing through points A and B can be found using the formula: slope = (y2 - y1) / (x2 - x1). In this case, the slope of AB is (3.7 - 1.0) / (6.1 - (-2)) = 0.6.
The perpendicular slope will be the negative reciprocal of the slope of AB, which is -1/0.6 = -5/3. This perpendicular line will pass through the midpoint of AB, which can be found using the midpoint formula: (x, y) = ((x1 + x2) / 2, (y1 + y2) / 2). Substituting the values, we get (x, y) = ((-2 + 6.1) / 2, (1 + 3.7) / 2) = (2.05, 2.35).
Now, we can use the point-slope form of the equation of a line: y - y1 = m(x - x1), where (x1, y1) is the midpoint and m is the slope. Substituting the values, we get y - 2.35 = (-5/3)(x - 2.05). We can solve this equation for y = c, the y-coordinate of point C:
y - 2.35 = (-5/3)(x - 2.05)
c - 2.35 = (-5/3)(1 - 2.05)
c - 2.35 = (-5/3)(-1.05)
c - 2.35 = 5/3 * 1.05
c - 2.35 = 35/60
c - 2.35 = 7/12
By adding 2.35 to both sides of the equation, we get:
c = 7/12 + 2.35
c = 0.5833 + 2.35
c = 2.9333
Therefore, the value of C that makes angle ABC a right angle is approximately 2.9333.