Solving Spatial Reasoning Problems with Coordinates

This problem asks for the distance between two points, X and Q, based on a series of relative positions and distances between several points (X, Y, Z, W, P, Q). To solve this, we can use coordinate geometry. We can assign coordinates to each point based on the given information and then use the distance formula.

Assigning Coordinates

Let's place one of the points at the origin (0,0) to make calculations easier. A good starting point might be Z or W. Let's place Z at the origin:

• Let Z be at (0, 0).

Now, let's determine the coordinates of the other points based on their descriptions relative to Z or other points whose coordinates we've found:

• W is 1 km west of Z. Moving west from (0,0) means decreasing the x-coordinate. So, W is at (-1, 0).
• Y is 1 km southeast of Z. Southeast is a diagonal direction. Moving 1 km southeast from (0,0) means moving equally in the positive x and negative y directions. The coordinates will be of the form (a, -a) such that the distance from (0,0) is 1 km. Using the distance formula, \(\sqrt{(a-0)^2 + (-a-0)^2} = 1\). This gives \(\sqrt{a^2 + a^2} = 1\), so \(\sqrt{2a^2} = 1\), which means \(a\sqrt{2} = 1\), and \(a = 1/\sqrt{2}\). So, Y is at \((1/\sqrt{2}, -1/\sqrt{2})\).
• X is 1 km northeast of Y. Northeast is also a diagonal direction. Moving 1 km northeast from Y means moving equally in the positive x and positive y directions relative to Y's position. From Y's coordinates \((1/\sqrt{2}, -1/\sqrt{2})\), moving 1 km northeast means adding \((1/\sqrt{2}, 1/\sqrt{2})\).
  • X's x-coordinate: \(1/\sqrt{2} + 1/\sqrt{2} = 2/\sqrt{2} = \sqrt{2}\).
  • X's y-coordinate: \(-1/\sqrt{2} + 1/\sqrt{2} = 0\).
  So, X is at \((\sqrt{2}, 0)\).
• P is 1 km south of W. W is at (-1, 0). Moving south from (-1,0) means decreasing the y-coordinate while keeping the x-coordinate the same. So, P is at (-1, 0 - 1) = (-1, -1).
• Q is 1 km east of P. P is at (-1, -1). Moving east from (-1,-1) means increasing the x-coordinate while keeping the y-coordinate the same. So, Q is at (-1 + 1, -1) = (0, -1).

Now we have the coordinates for X and Q:

• X: \((\sqrt{2}, 0)\)
• Q: (0, -1)

Calculating the Distance Between X and Q

To find the distance between X\((\sqrt{2}, 0)\) and Q(0, -1), we use the distance formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) in a 2D plane:

Distance \(= \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}\)

Let \((x_1, y_1) = (\sqrt{2}, 0)\) and \((x_2, y_2) = (0, -1)\). Plugging these values into the formula:

Distance XQ \(= \sqrt{((0 - \sqrt{2})^2 + (-1 - 0)^2)}\)

Distance XQ \(= \sqrt{( (-\sqrt{2})^2 + (-1)^2 )}\)

Distance XQ \(= \sqrt{(2 + 1)}\)

Distance XQ \(= \sqrt{3}\)

The distance between X and Q is \(\sqrt{3}\) km.

Summary of Point Coordinates

Revision Table: Key Concepts

Additional Information on Spatial Reasoning and Coordinates

Spatial reasoning problems often require visualizing positions and movements. Using a coordinate system is a powerful method to solve these problems systematically, especially when precise distances are involved. By setting up a coordinate system, you convert directional information into numerical changes in x and y values. This allows you to use algebraic and geometric formulas, like the distance formula, to find solutions.

When dealing with diagonal directions (like northeast, southeast), remember that a movement of distance 'd' in a perfect diagonal (45 degrees to axes) corresponds to a change of \(d/\sqrt{2}\) in both the x and y coordinates, with signs depending on the quadrant.

Practicing converting directional descriptions into coordinate changes will improve your ability to solve these types of questions efficiently.