Triangular Park Path and Fountain Circle on a Coordinate Map
A city planning map uses a coordinate grid measured in hundreds of feet. A triangular park has vertices A(-4, 1), B(6, 3), and C(2, 9). The parks department proposes a straight footpath along the line y = -x + 5, claiming it is the perpendicular bisector of AB and therefore every point on the path is equally far from A and B. A donor will fund a circular fountain centered at the point equidistant from all three vertices only if that center lies inside the triangle. The department also needs the equation of the circle through A, B, and C to check whether a proposed statue at P(0, 3) would fall inside the fountain's circular basin. The map is fixed; no new survey points can be added. The staff needs to know whether the footpath claim is correct, where the fountain center is, and whether the statue location is inside the circle.
1. Is the proposed path y = -x + 5 the perpendicular bisector of AB? Check both the midpoint and the slope.
Read the worked answer
The midpoint of AB is M = ((-4 + 6)/2, (1 + 3)/2) = (1, 2).
The slope of AB is m_AB = (3 − 1)/(6 − (-4)) = 2/10 = 1/5.
A perpendicular bisector must pass through M and have slope −5. Its equation is y − 2 = −5(x − 1), or y = −5x + 7.
The proposed line y = −x + 5 has slope −1, not −5. Also, at x = 1 it gives y = 4, not the midpoint y = 2. So it is neither perpendicular to AB nor a bisector of AB, and points on it are not generally equidistant from A and B.
2. Find the circumcenter of triangle ABC, verify that it is inside the triangle, and write the equation of the circle through A, B, and C.
Read the worked answer
Use two perpendicular bisectors. The perpendicular bisector of AB is y = −5x + 7 from the previous part.
For BC, the midpoint is ((6 + 2)/2, (3 + 9)/2) = (4, 6). The slope of BC is (9 − 3)/(2 − 6) = 6/(-4) = −3/2, so the perpendicular slope is 2/3. Thus
y − 6 = (2/3)(x − 4) y = (2/3)x + 10/3.
Intersect with y = −5x + 7: −5x + 7 = (2/3)x + 10/3 −15x + 21 = 2x + 10 11 = 17x x = 11/17. Then y = −5(11/17) + 7 = 64/17.
The circumcenter is (11/17, 64/17) ≈ (0.65, 3.76).
The squared radius is the squared distance to A(-4, 1): (-4 − 11/17)² + (1 − 64/17)² = (-79/17)² + (-47/17)² = (6241 + 2209)/289 = 8450/289 ≈ 29.24. So r ≈ 5.41.
Circle: (x − 11/17)² + (y − 64/17)² = 8450/289.
To verify the center is inside, note the side squares are AB² = 104, BC² = 52, and CA² = 100. The largest, 104, is less than 52 + 100 = 152, so the triangle is acute, and the circumcenter of an acute triangle lies inside it.
3. Is the statue point P(0, 3) inside the fountain circle? Support your answer with a distance comparison.
Read the worked answer
Compare the squared distance from P to the circumcenter with the squared radius.
Distance squared = (0 − 11/17)² + (3 − 64/17)² = (121/289) + ((51 − 64)/17)² = 121/289 + 169/289 = 290/289 ≈ 1.00.
The squared radius is 8450/289 ≈ 29.24. Since 290/289 < 8450/289, P is much closer to the center than the radius, so P is inside the circle. The statue location is acceptable for the circular basin as described.