Write a system of inequalities to express the exterior of the triangle ABD if A=(-2,7), B=(4,0), and C=(1,-3).
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Derive equations for the three lines connecting the three points. Then change the equations to inequalities to represent the region on the side of the lines that is outside the triangle.
It will help if you draw the three points on graph paper and connect them with extended lines passing through pairs of points.
Example: The line connecting A and B has slope m = (7-0)/(-2-4) = -7/6 and the equation for this line, extended to infinity, is
(y-7) = (-7/6) (x+2) or
y = -7x/6 -7/3 + 7 = -7x/6 +14/3
The region above this line is outside the triangle, so one of the inequalities is
y > -7x/6 +14/3
To express the exterior of the triangle ABD, we need to derive equations for the three lines that connect the points A, B, and D. Let's start by finding the equation of the line passing through points A and B.
1. Equation of the line AB:
The slope of the line AB can be found using the formula: m = (y2 - y1) / (x2 - x1) = (0 - 7) / (4 - (-2)) = -7/6
Using the slope-intercept form (y = mx + b), we substitute the slope and the coordinates of point A into the equation:
y - 7 = (-7/6)(x - (-2))
y - 7 = (-7/6)(x + 2)
y - 7 = (-7/6)x - 7/3
y = (-7/6)x + 14/3
Now, we need to represent the region outside the triangle on the side of the line AB. Since the triangle ABD is below the line AB, the inequality will be:
y < (-7/6)x + 14/3
Next, let's find the equation of the line passing through points A and D.
2. Equation of the line AD:
The slope of the line AD can be found using the formula: m = (y2 - y1) / (x2 - x1) = (-3 - 7) / (1 - (-2)) = -10/3
Using the slope-intercept form, we substitute the slope and the coordinates of point A into the equation:
y - 7 = (-10/3)(x - (-2))
y - 7 = (-10/3)(x + 2)
y - 7 = (-10/3)x - 20/3
y = (-10/3)x + 41/3
Since the triangle ABD is on the right side of the line AD, the inequality will be:
x > -2
Finally, let's find the equation of the line passing through points B and D.
3. Equation of the line BD:
The slope of the line BD can be found using the formula: m = (y2 - y1) / (x2 - x1) = (-3 - 0) / (1 - 4) = -3/(-3) = 1
Using the slope-intercept form, we substitute the slope and the coordinates of point B into the equation:
y - 0 = 1(x - 4)
y = x - 4
Since the triangle ABD is on the left side of the line BD, the inequality will be:
x < 4
Putting it all together, the system of inequalities to express the exterior of triangle ABD is:
y < (-7/6)x + 14/3
x > -2
x < 4