Given the points A(−2, 1) and B(3, 4), what are the coordinates of point C in the fourth
quadrant such that mCAB = 90 degrees and AB = AC? Express your answer as an ordered pair.
To find the coordinates of point C, we need to follow these steps:
1. Calculate the midpoint of points A and B.
2. Calculate the slope of the line passing through A and B.
3. Calculate the negative reciprocal of the slope from step 2 to find the slope of a line perpendicular to AB.
4. Use the midpoint from step 1 and the perpendicular slope from step 3 to find the equation of the line passing through the midpoint.
5. Find the point where the line from step 4 intersects the x-axis to get the x-coordinate of point C.
6. Substitute the x-coordinate from step 5 into the equation of the line from step 4 to find the y-coordinate of point C.
Let's go through these steps:
1. The midpoint of points A(-2, 1) and B(3, 4) can be calculated by finding the average of their x-coordinates and y-coordinates:
Midpoint = ((-2 + 3) / 2, (1 + 4) / 2)
= (1/2, 5/2)
2. The slope of the line passing through points A and B can be calculated using the formula:
Slope = (y2 - y1) / (x2 - x1)
= (4 - 1) / (3 - (-2))
= 3/5
3. The negative reciprocal of the slope from step 2 can be obtained by flipping the fraction and changing the sign:
Negative Reciprocal Slope = -5/3
4. We can now find the equation of the line passing through the midpoint (1/2, 5/2) with the slope -5/3 using the point-slope form:
y - y1 = m(x - x1)
y - 5/2 = -5/3(x - 1/2)
y - 5/2 = -5/3x + 5/6
y = -5/3x + 5/6 + 5/2
y = -5/3x + 25/6
5. To find the x-coordinate of point C, we need to find the point where the line intersects the x-axis. This occurs when y = 0:
0 = -5/3x + 25/6
-5/3x = -25/6
x = (-25/6) / (-5/3)
x = 5/2
6. Finally, we substitute the x-coordinate, x = 5/2, into the equation of the line from step 4 to find the y-coordinate of point C:
y = -5/3(5/2) + 25/6
y = -25/6 + 25/6
y = 0
Therefore, the coordinates of point C in the fourth quadrant such that m∠CAB = 90 degrees and AB = AC are (5/2, 0).