Question 1: Plot the locus of points equidistant from points A(2,3) and B(6,7). Then, plot the locus of points equidistant from points C(4,1) and D(8,5). Identify the region formed by the intersection of these two loci.

Answer: The intersection region is a lens-shaped area between the perpendicular bisectors of AB and CD.

Question 2: Construct the locus of points that are equidistant from the circle centered at (3,3) with radius 2 and the point (6,6). Use the grid to illustrate your construction.

Answer: The locus is the perpendicular bisector of the segment connecting the circle's center and the point, forming a line on the grid.

Question 3: A point P is such that its distance from A(1,2) is always equal to its distance from the locus of points equidistant from B(4,5) and C(7,8). Plot these loci and describe the region where point P can lie.

Answer: Point P lies on the circle centered at A with a radius equal to the distance from A to the locus line, which is the perpendicular bisector of BC.

Question 4: Determine the region formed by combining the loci of points equidistant from F(2,4), G(6,4), and H(4,8). Use the grid to draw the combined locus.

Answer: The region is the intersection of three regions: the perpendicular bisectors of FG and FH, forming a triangular-shaped area.

Question 5: A student constructs two loci: one is the circle centered at (5,5) with radius 3, and the other is the parabola y=x^2 - 4x + 5. Plot both and identify their intersection points.

Answer: Intersections are approximately at (2,1) and (8,1).

Question 6: Explain how the intersection of two loci can be used to find the solution to a real-world problem involving navigation or positioning.

Answer: Intersecting loci can identify points that satisfy multiple conditions simultaneously, such as locations equidistant from two landmarks, aiding in precise positioning.

Question 7: A building site requires placing a water tank such that it is equidistant from two existing trees located at (2,3) and (8,3). Using the grid, construct this locus and suggest an optimal placement on the site.

Answer: The locus is the perpendicular bisector of the segment joining the two trees, forming a vertical line at x=5.

Question 8: Construct the locus of points that are equidistant from the point E(4,2) and the line y=6. Use the grid to illustrate.

Answer: The locus is the horizontal line y=4, midway between E and the line y=6.

Question 9: Identify and correct the mistake in the following construction: A student draws the locus of points equidistant from (1,1) and (5,5) but mistakenly plots it as a circle centered at (3,3) with radius 2.

Answer: The correct locus is the perpendicular bisector of the segment connecting (1,1) and (5,5), which is the line y=x, not a circle.

Question 10: Challenge: Using the grid, construct the loci of points equidistant from three points A(1,1), B(7,1), and C(4,7). Describe the region where all three loci intersect.

Answer: The intersection region is a single point—the triangle's incenter—located at approximately (4,3).

Question 11: Mixed Review: Plot the locus of points that are 4 units away from point D(3,4) and also lie on the locus of points equidistant from points E(6,4) and F(3,7). Describe the combined region.

Answer: The locus of points 4 units from D is a circle centered at D with radius 4; the other locus is the perpendicular bisector of EF. The region is their intersection, which is a segment of the circle near the bisector.