To solve this, we need to use the information given about the relationship between the diagonal and the sides of the rectangle and the length of one side (70 cm) to calculate the perimeter.

Step 1: Relationship of the diagonal and sides

The diagonal ($d$) and one side ($a$) of the rectangle are given in the ratio $37 : 12$. This means:

$\frac{d}{a} = \frac{37}{12}$

or equivalently,

$d = \frac{37}{12} \cdot a$

Step 2: Use the Pythagorean theorem

In a rectangle, the diagonal, the longer side ($b$), and the shorter side ($a$) are related by the Pythagorean theorem:

$d^2 = a^2 + b^2$

Substitute $d = \frac{37}{12} \cdot a$ into this equation:

$\left(\frac{37}{12} \cdot a\right)^2 = a^2 + b^2$

Simplify:

$\frac{1369}{144} \cdot a^2 = a^2 + b^2$

Multiply through by 144 to eliminate the fraction:

$1369 \cdot a^2 = 144 \cdot a^2 + 144 \cdot b^2$

Simplify further:

$1369a^2 - 144a^2 = 144b^2$

$1225a^2 = 144b^2$

$\frac{a^2}{b^2} = \frac{144}{1225}$

$\frac{a}{b} = \frac{12}{35}$

This gives the ratio of the shorter side to the longer side as $12 : 35$.

Step 3: Find the shorter side

We are given the longer side ($b$) as $70 \, \text{cm}$. Using the ratio $a : b = 12 : 35$, we can calculate the shorter side:

$a = \frac{12}{35} \cdot b$

$a = \frac{12}{35} \cdot 70 = 24 \, \text{cm}$

So, the shorter side ($a$) is $24 \, \text{cm}$.

Step 4: Calculate the perimeter

The perimeter of a rectangle is given by:

$P = 2(a + b)$

Substitute $a = 24 \, \text{cm}$ and $b = 70 \, \text{cm}$:

$P = 2(24 + 70)$

$P = 2 \cdot 94 = 188 \, \text{cm}$

Final Answer:

The perimeter of the rectangle is 188 cm.

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Would you like a detailed breakdown of any step?
Here are 5 related questions to explore further:

1. How is the Pythagorean theorem applied to calculate the diagonal of a rectangle?
2. Can you explain why the ratio $37:12$ was important in solving this problem?
3. How do ratios help in solving real-world geometry problems?
4. What happens if the diagonal-to-side ratio changes?
5. How can this solution be applied to find the area of the rectangle?

Tip: Always double-check your ratios and substitutions when working with proportional relationships!