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Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is
- a)1/2
- b)2/3
- c)1/4
- d)3/4
Correct answer is option 'D'. Can you explain this answer?
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Given:
- A rectangular field
- The boy took a shortcut along the diagonal
- The boy saved a distance equal to half the longer side
To Find:
The ratio of the shorter side to the longer side
Solution:
Step 1: Understanding the Problem
Let's assume the shorter side of the rectangular field is 'a' units and the longer side is 'b' units. We need to find the ratio of the shorter side to the longer side, which is a/b.
Step 2: Visualizing the Problem
To better understand the problem, let's draw a rectangular field and label its sides.
```
________ b ________
| |
a | |
| |
-------------------
```
Step 3: Analyzing the Shortcut
When the boy takes a shortcut along the diagonal, he saves a distance equal to half the longer side. This means that the length of the diagonal is equal to (b/2) units.
Step 4: Applying the Pythagorean Theorem
In a right-angled triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
Using the Pythagorean theorem, we can write the equation as:
a^2 + b^2 = (b/2)^2
Step 5: Solving the Equation
Let's solve the equation to find the relationship between a and b.
Expanding the equation:
a^2 + b^2 = b^2/4
Multiplying both sides of the equation by 4:
4a^2 + 4b^2 = b^2
Subtracting b^2 from both sides of the equation:
4a^2 = -3b^2
Dividing both sides of the equation by b^2:
4(a^2/b^2) = -3
Simplifying:
a^2/b^2 = -3/4
Taking the square root of both sides:
a/b = √(-3/4)
Since the square root of a negative number is not defined in the real number system, we can conclude that the ratio a/b does not exist.
Step 6: Conclusion
Based on the analysis, it is impossible to determine the ratio of the shorter side to the longer side. Therefore, none of the given options (a, b, c, d) is the correct answer.
Note: It is important to double-check the given question and options to ensure their accuracy.
- A rectangular field
- The boy took a shortcut along the diagonal
- The boy saved a distance equal to half the longer side
To Find:
The ratio of the shorter side to the longer side
Solution:
Step 1: Understanding the Problem
Let's assume the shorter side of the rectangular field is 'a' units and the longer side is 'b' units. We need to find the ratio of the shorter side to the longer side, which is a/b.
Step 2: Visualizing the Problem
To better understand the problem, let's draw a rectangular field and label its sides.
```
________ b ________
| |
a | |
| |
-------------------
```
Step 3: Analyzing the Shortcut
When the boy takes a shortcut along the diagonal, he saves a distance equal to half the longer side. This means that the length of the diagonal is equal to (b/2) units.
Step 4: Applying the Pythagorean Theorem
In a right-angled triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
Using the Pythagorean theorem, we can write the equation as:
a^2 + b^2 = (b/2)^2
Step 5: Solving the Equation
Let's solve the equation to find the relationship between a and b.
Expanding the equation:
a^2 + b^2 = b^2/4
Multiplying both sides of the equation by 4:
4a^2 + 4b^2 = b^2
Subtracting b^2 from both sides of the equation:
4a^2 = -3b^2
Dividing both sides of the equation by b^2:
4(a^2/b^2) = -3
Simplifying:
a^2/b^2 = -3/4
Taking the square root of both sides:
a/b = √(-3/4)
Since the square root of a negative number is not defined in the real number system, we can conclude that the ratio a/b does not exist.
Step 6: Conclusion
Based on the analysis, it is impossible to determine the ratio of the shorter side to the longer side. Therefore, none of the given options (a, b, c, d) is the correct answer.
Note: It is important to double-check the given question and options to ensure their accuracy.
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Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side isa)1/2b)2/3c)1/4d)3/4Correct answer is option 'D'. Can you explain this answer?
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