Manjot cycles X km towards North and then turns left and cycles Y km to reach her shop. The shortest distance between the starting point and her shop is 5 km. What can be the value of X and Y in km?
3 and 4
Understanding the Problem: Distance and Displacement
The question describes Manjot's journey in two parts: first, cycling North, and second, turning Left and cycling West. The key is to find the shortest distance between her starting point and the final destination (her shop). The movement towards North and then West creates a specific geometric shape.
When someone moves in one direction (North) and then turns 90 degrees (Left from North is West) and moves in another direction, their path forms two sides of a right-angled triangle. The starting point, the turning point, and the final point form the vertices of this triangle. The distance covered North (X km) is one leg, and the distance covered West (Y km) is the other leg.
The shortest distance between the starting point and the ending point is the straight line connecting these two points. In a right-angled triangle, this straight line is the hypotenuse.
Applying the Pythagorean Theorem for Shortest Distance
The relationship between the sides of a right-angled triangle is described by the Pythagorean theorem. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).
Let:
- The distance cycled North be X km.
- The distance cycled West be Y km.
- The shortest distance between the start and the shop be H km.
According to the problem, H = 5 km. The Northward distance (X) and the Westward distance (Y) are the legs of the right-angled triangle, and the shortest distance (5 km) is the hypotenuse.
The Pythagorean theorem can be written as:
Leg\(1)^2\) + Leg\(2)^2\) = Hypotenuse\(^2\)
Substituting the values from the problem:
\(X^2 + Y^2 = 5^2\)
This simplifies to:
\(X^2 + Y^2 = 25\)
We need to find which pair of values for X and Y from the options satisfies this equation.
Evaluating the Options for X and Y Values
Let's test each option to see if \(X^2 + Y^2\) equals 25 km\(^2\).
| Option | Value of X (km) | Value of Y (km) | Calculation \(X^2 + Y^2\) | Result | Does it equal 25? |
|---|---|---|---|---|---|
| 1 | 3 | 4 | \(3^2 + 4^2 = 9 + 16 = 25\) | 25 | Yes |
| 2 | 8 | 4 | \(8^2 + 4^2 = 64 + 16 = 80\) | 80 | No |
| 3 | 3 | 5 | \(3^2 + 5^2 = 9 + 25 = 34\) | 34 | No |
| 4 | 4 | 6 | \(4^2 + 6^2 = 16 + 36 = 52\) | 52 | No |
From the evaluation, only the values X = 3 km and Y = 4 km satisfy the equation \(X^2 + Y^2 = 25\). Therefore, these are the possible values for the distances cycled North and West.
Conclusion: Determining the Correct X and Y
The journey described creates a right-angled triangle where the distances cycled (X and Y) are the legs and the shortest distance (5 km) is the hypotenuse. Using the Pythagorean theorem, \(X^2 + Y^2 = 5^2\). Checking the options reveals that only X=3 and Y=4 satisfy this condition \(3^2 + 4^2 = 9 + 16 = 25\).
Revision Table: Key Concepts
| Concept | Description | Application in this Problem |
|---|---|---|
| Direction | North, Left (which is West relative to North) | Defines the path taken by Manjot. |
| Displacement | The shortest distance between starting and ending points. | Given as 5 km, this is the hypotenuse. |
| Right-Angled Triangle | A triangle with one 90-degree angle. | Formed by the Northward path, Westward path, and the shortest distance. |
| Pythagorean Theorem | \(a^2 + b^2 = c^2\) for a right triangle legs a, b and hypotenuse c. | Used to relate X, Y, and the 5 km shortest distance: \(X^2 + Y^2 = 5^2\). |
Additional Information: Pythagorean Triples
The numbers 3, 4, and 5 form a set known as a Pythagorean triple. A Pythagorean triple consists of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\). Common Pythagorean triples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and their multiples (e.g., 6, 8, 10 is a multiple of 3, 4, 5, since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). Recognizing these triples can quickly help solve problems involving right-angled triangles and the Pythagorean theorem, such as finding the shortest distance or displacement in geometry and physics problems.
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Rohan walks 10 km towards north from a point A. Then he turns right and walks 6 km. Then, he again turns right and walks 10 km. In which direction is he now from the starting point?
