Question

Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved the distance equal to half of the longer side. Then the ratio of the shorter side to the longer side is

Answer 1

Correct Answer: 2

Step 1: Understanding the Question:
The problem requires us to find the ratio of the shorter side to the longer side of a specific rectangular field.
A boy originally plans to walk along two adjacent sides of this rectangle.
Instead, he takes a shortcut straight across the diagonal of the field.
By doing this, the distance he saves is exactly equal to half the length of the longer side of the rectangle.
Step 2: Key Formula or Approach:
For a rectangle with sides $a$ and $b$, the standard path along the two adjacent sides has a length of $a + b$.
The diagonal path, which forms the hypotenuse of a right-angled triangle, has a length of $\sqrt{a^2 + b^2}$ according to Pythagoras' theorem.
The saved distance is the difference between these two paths.
We will set up an algebraic equation based on the condition given and solve for the ratio.
Step 3: Detailed Explanation:

Let the longer side of the rectangular field be denoted by $a$.

Let the shorter side of the rectangular field be denoted by $b$.

The distance covered by walking along the two adjacent sides is $a + b$.

The distance covered by taking the diagonal shortcut is $\sqrt{a^2 + b^2}$.

The amount of distance saved is the difference between the standard path and the shortcut path.

Distance saved = $(a + b) - \sqrt{a^2 + b^2}$.

The problem states that this saved distance is equal to exactly half of the longer side, which is $\frac{a}{2}$.

We can express this relationship mathematically as: \[ a + b - \sqrt{a^2 + b^2} = \frac{a}{2} \]

We will rearrange this equation to isolate the square root on one side.

\[ a + b - \frac{a}{2} = \sqrt{a^2 + b^2} \]

Simplifying the left side gives: \[ \frac{a}{2} + b = \sqrt{a^2 + b^2} \]

To eliminate the square root, we square both sides of the equation.

\[ \left( \frac{a}{2} + b \right)^2 = \left( \sqrt{a^2 + b^2} \right)^2 \]

We expand the left side using the algebraic identity $(x + y)^2 = x^2 + 2xy + y^2$.

\[ \frac{a^2}{4} + 2\left(\frac{a}{2}\right)(b) + b^2 = a^2 + b^2 \]

This simplifies to: \[ \frac{a^2}{4} + ab + b^2 = a^2 + b^2 \]

We subtract $b^2$ from both sides, effectively cancelling it out completely.

\[ \frac{a^2}{4} + ab = a^2 \]

Next, we isolate the $ab$ term by subtracting $\frac{a^2}{4}$ from $a^2$.

\[ ab = a^2 - \frac{a^2}{4} \]

\[ ab = \frac{3a^2}{4} \]

Since $a$ represents a physical length, it cannot be zero.

Therefore, we can safely divide both sides of the equation by $a$.

\[ b = \frac{3a}{4} \]

To find the required ratio of the shorter side to the longer side, we divide by $a$.

\[ \frac{b}{a} = \frac{3}{4} \]

The ratio of the shorter side to the longer side is exactly 3/4.

Step 4: Final Answer:
The ratio of the shorter side to the longer side is 3/4.