Question : The length of the diagonals of a rhombus is 40 cm and 60 cm. What is the length of the side of the rhombus?
Option 1: $50 \sqrt{3} \ \text{cm}$
Option 2: $20 \sqrt{3}\ \text{cm}$
Option 3: $10 \sqrt{13}\ \text{cm}$
Option 4: $40 \sqrt{13}\ \text{cm}$
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Correct Answer: $10 \sqrt{13}\ \text{cm}$
Solution :
Let's denote the length of the side of the rhombus as $s$ and the diagonal as $d1$ and $d2$.
According to the given information, the diagonals of the rhombus are 40 cm and 60 cm.
These diagonals divide the rhombus into four congruent right-angled triangles.
Using the Pythagorean theorem, we can write the following equation for one of the right-angled triangles:
$(\frac{d1}{2})^2 + (\frac{d2}{2})^2 = (s)^2$
Simplifying this equation:
⇒ $(\frac{40}{2})^2 + (\frac{60}{2})^2 = (s)^2$
⇒ $(20)^2 + (30)^2 = (s)^2$
⇒ $s^2 = 1300$
⇒ $s = \sqrt{1300}$
⇒ $s = 10\sqrt{13}\ \text{cm}$
Hence, the correct answer is $10\sqrt13\ \text{cm}$.
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