Direction & Distance Aptitude

Cheatsheet Content

### Basics of Directions - **Cardinal Directions:** North (N), South (S), East (E), West (W). - **Ordinal Directions (Sub-directions):** - North-East (NE): Between N and E - North-West (NW): Between N and W - South-East (SE): Between S and E - South-West (SW): Between S and W - **Reference Frame:** Always assume you are facing North unless stated otherwise. - **Clockwise/Anti-clockwise:** - Right turn = 90° clockwise - Left turn = 90° anti-clockwise - **Angle Turns:** A turn is usually 90 degrees unless specified (e.g., "turns 45 degrees left"). ### Pythagorean Theorem - **Concept:** Used to find the shortest distance between two points when movements are perpendicular (forming a right-angled triangle). - **Formula:** $H^2 = P^2 + B^2$ - H = Hypotenuse (shortest distance) - P = Perpendicular (vertical distance) - B = Base (horizontal distance) - **Common Pythagorean Triplets:** (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Multiples also apply (e.g., 6, 8, 10). ### Type 1: Single Person Movement - **Description:** Questions describe the movement of a single person or object from a starting point, asking for final direction or distance from the start. - **Strategy:** 1. **Draw a Diagram:** Represent each movement with an arrow. 2. **Break Down:** Separate horizontal (East-West) and vertical (North-South) movements. 3. **Net Displacement:** Calculate net East/West and net North/South displacement. 4. **Final Direction:** Determine the final direction relative to the starting point. 5. **Final Distance:** Use Pythagorean theorem if movements form a right triangle, or simply sum/subtract if movements are collinear. #### Example: A person walks 5 km North, then turns right and walks 3 km, then turns left and walks 2 km. What is their final direction from the starting point? - **Step 1:** Draw. - **Step 2:** - North: +5 km - East: +3 km (after right turn) - North: +2 km (after left turn) - **Step 3:** - Net North = 5 + 2 = 7 km - Net East = 3 km - **Step 4:** Final Direction from start = North-East. ### Type 2: Multiple Persons / Relative Directions - **Description:** Involves two or more people/objects moving, and questions ask about their relative positions or directions from each other. - **Strategy:** 1. **Draw Separate Paths:** Clearly mark the starting and ending points for each person. 2. **Combine Diagrams (if necessary):** Place starting points at a common origin if they start from the same place. 3. **Identify Relative Position:** To find A's direction from B, imagine standing at B and looking towards A. 4. **Calculate Relative Distance:** Use Pythagorean theorem or direct summation/subtraction based on the combined diagram. #### Example: Person A walks 4 km East from point O. Person B walks 3 km North from point O. What is the shortest distance between A and B, and what is B's direction from A? - **Step 1 & 2:** Both start at O. A is 4 km East of O. B is 3 km North of O. - **Step 3:** From A, B is North-West. - **Step 4:** A right triangle is formed with sides 3 km (N) and 4 km (E). - Shortest distance (Hypotenuse) = $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ km. ### Type 3: Shadow-Based Problems - **Description:** These problems involve shadows, usually during sunrise or sunset, and ask for the direction a person is facing. - **Key Concepts:** - **Sunrise (Morning):** Sun is in the East. Shadow falls to the West. - **Sunset (Evening):** Sun is in the West. Shadow falls to the East. - **Noon:** No shadow (or directly beneath). - **Person's Left/Right:** If a shadow falls to a person's left, their right hand is towards the shadow's direction. - **Strategy:** 1. **Identify Time of Day:** Sunrise or Sunset. This determines the sun's position and general shadow direction. 2. **Determine Shadow Direction:** Based on sun's position. 3. **Relate Shadow to Person's Body:** If shadow is to the left/right, align the person's body accordingly. 4. **Deduce Facing Direction:** Once the person's left/right is oriented, their facing direction becomes clear. #### Example: One morning, Ram and Shyam were talking face to face. If Ram's shadow was exactly to his left, which direction was Shyam facing? - **Step 1:** Morning, so Sun is in the East. - **Step 2:** Shadow falls to the West. - **Step 3:** Ram's shadow is to his left. So, Ram's left is West. - **Step 4:** If Ram's left is West, then Ram is facing South. Since they are face to face, Shyam must be facing North. ### Type 4: Angle-Based Turns - **Description:** Instead of "left" or "right," turns are specified in degrees (e.g., 45°, 135°, 270°). - **Strategy:** 1. **Initial Direction:** Start with the given initial direction. 2. **Convert Turns to Degrees:** - Right turn = Clockwise (+) - Left turn = Anti-clockwise (-) 3. **Cumulative Angle:** Sum up all clockwise and anti-clockwise turns. 4. **Determine Final Direction:** - Divide the net angle by 90 to see how many 90° turns it represents. - Each 90° clockwise turn moves you one cardinal direction (N -> E -> S -> W -> N). - Each 90° anti-clockwise turn moves you one cardinal direction (N -> W -> S -> E -> N). - Remember 360° brings you back to the start. #### Example: A person is facing North. They turn 45° clockwise, then 135° anti-clockwise, then 270° clockwise. What direction are they facing? - **Step 1:** Starts North. - **Step 2 & 3:** - +45° (clockwise) - -135° (anti-clockwise) - +270° (clockwise) - Net angle = 45 - 135 + 270 = 180° clockwise. - **Step 4:** From North, 180° clockwise means facing South. ### Type 5: Coding-Decoding Directions - **Description:** Directions are assigned codes (e.g., 'A' means North, 'B' means East). You need to decode and solve. - **Strategy:** 1. **Decode the Codes:** Write down the actual direction for each code given. 2. **Translate the Problem:** Rewrite the problem using the decoded directions. 3. **Solve as Standard Problem:** Apply strategies from Type 1 or 2. #### Example: If 'A' means North, 'B' means East, 'C' means South, 'D' means West. A person walks 5 km A, then turns right and walks 3 km B. What is their final direction from the starting point? - **Step 1:** A = North, B = East. - **Step 2:** "A person walks 5 km North, then turns right and walks 3 km East." - **Step 3:** (Solve as Type 1) - Net North = 5 km - Net East = 3 km - Final Direction = North-East. ### Key Tips & Common Mistakes - **Draw Diagrams:** Always draw a clear diagram. It's the most crucial step. - **Cardinal Points:** Clearly label N, S, E, W on your diagram. - **Right/Left Turns:** - Facing North: Right = East, Left = West - Facing East: Right = South, Left = North - Facing South: Right = West, Left = East - Facing West: Right = North, Left = South - **"From" vs. "To":** Pay attention to whether the question asks for direction *from* the start *to* the end, or *from* the end *to* the start. - **Shortest Distance:** Usually implies a straight line, often requiring the Pythagorean theorem. - **Total Distance:** Sum of all individual path segments. - **Cumulative Turns:** For angle-based problems, sum up all turns (clockwise positive, anti-clockwise negative) to find the net turn. - **Relative Direction:** To find direction of A from B, place yourself at B and look towards A. - **Shadow Problems:** Always fix the sun's position first (East for morning, West for evening).