Case Study 1: Identify the quadrants for the points $A(3, -4)$, $B(-2, -5)$, $C(0, 4)$, and $D(-1, 2)$. Explain your reasoning based on the signs of the coordinates.

Case Study 2: A point $P(4, 7)$ is reflected across the x-axis to form point $P'$. Then, $P'$ is reflected across the y-axis to form point $P''$. Find the coordinates of $P'$ and $P''$.

Case Study 1: Calculate the directed distance from point $A(-3, 5)$ to point $B(8, 5)$. What does the sign of the result indicate?

Case Study 2: Calculate the directed distance from point $C(2, 6)$ to point $D(2, -4)$. What does the sign of the result indicate?

Problem 1: Find the absolute distance between points $P(2, -3)$ and $Q(-4, 5)$.

Problem 2: The distance between the points $A(x, 2)$ and $B(3, -6)$ is $10$ units. Find the possible values for $x$.

Problem 3: Verify that the points $A(1, 1)$, $B(4, 5)$, and $C(1, 5)$ form a right-angled triangle.

Problem 1: Find the midpoint of the segment connecting $A(-5, 8)$ and $B(3, -2)$.

Problem 2: The midpoint of a line segment is $M(4, -1)$. If one endpoint is $P_1(1, 5)$, find the coordinates of the other endpoint $P_2(x, y)$.

Problem 3: Find the center of a circle whose diameter has endpoints $A(-3, 4)$ and $B(9, -2)$.

Problem 1: Find the point dividing the segment $P_1(1, 4)$ and $P_2(6, 9)$ internally in the ratio $2:3$.

Problem 2: Find the coordinates of the point that divides the line segment joining $A(-2, 3)$ and $B(4, -1)$ externally in the ratio $5:2$.

Problem 3: In what ratio does the x-axis divide the line segment connecting points $A(2, -3)$ and $B(5, 6)$?

Problem 1: Find the slope and inclination angle of the line passing through points $A(1, 2)$ and $B(4, 5)$.

Problem 2: Determine if the points $P(-2, -4)$, $Q(1, 2)$, and $R(3, 6)$ are collinear.

Problem 3: A line has an inclination of $135^\circ$ and passes through the point $(2, 3)$. Find the y-coordinate of a point on this line if its x-coordinate is $5$.

Problem 1: Calculate the area of a triangle with vertices $A(1, 2)$, $B(4, 6)$, and $C(7, 3)$.

Problem 2: Find the area of the quadrilateral with vertices $(-2, 3)$, $(3, 5)$, $(4, -1)$, and $(0, -4)$ listed consecutively.

Problem 3: The area of a triangle formed by points $(1, 1)$, $(5, 1)$, and $(x, 4)$ is $10$ square units. Find the possible values for $x$.

Problem 1: Identify the type of conic section represented by the equation $4x^2 - 9y^2 + 32x + 54y - 100 = 0$.

Problem 2: Determine the conic section defined by $x^2 + y^2 - 6x + 8y - 24 = 0$.

Problem 3: Identify the graph of $y^2 - 4y - 8x + 28 = 0$.

Case Study 1: Prove analytically that the diagonals of a rectangle are equal in length.

Case Study 2: Prove analytically that the line segment joining the midpoints of two sides of a triangle is parallel to the third side.