The area of a rectangle and the square of its perimeter are in the ratio $1:25$. Then the lengths of the shorter and longer sides of the rectangle are in the ratio

We're told that for a rectangle:

Area : (Perimeter)² = 1 : 25

We need to find the ratio of the shorter side to the longer side.

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Let's call the sides of the rectangle $a$ and $b$.

Area of rectangle = $a \times b$

Perimeter of rectangle = $2(a + b)$

Square of perimeter = $[2(a + b)]^2$

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From the given ratio: Area : (Perimeter)² = 1 : 25

This means: $\frac{\text{Area}}{(\text{Perimeter})^2} = \frac{1}{25}$

$\frac{ab}{[2(a + b)]^2} = \frac{1}{25}$

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$25ab = [2(a + b)]^2$

$25ab = 4(a + b)^2$

$25ab = 4(a^2 + 2ab + b^2)$

$25ab = 4a^2 + 8ab + 4b^2$

$25ab - 8ab = 4a^2 + 4b^2$

$17ab = 4(a^2 + b^2)$

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Dividing both sides by $ab$:

$17 = \frac{4(a^2 + b^2)}{ab}$

$17 = 4\left(\frac{a}{b} + \frac{b}{a}\right)$

Let $r = \frac{a}{b}$ (ratio of one side to the other):

$17 = 4\left(r + \frac{1}{r}\right)$

$\frac{17}{4} = r + \frac{1}{r}$

$4.25 = r + \frac{1}{r}$

This gives us: $r^2 - 4.25r + 1 = 0$

Using the quadratic formula: $r = 4$ or $r = \frac{1}{4}$

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Since we want the ratio of shorter : longer side:

If $r = 4$, then one side is 4 times the other, so shorter : longer = 1 : 4

If $r = \frac{1}{4}$, then one side is $\frac{1}{4}$ of the other, so shorter : longer = 1 : 4

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Therefore, the ratio of shorter to longer sides is 1 : 4