A right-angled triangle has a hypotenuse o... | গণিত Multiple Choice Question

<html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>Right-Angled Triangle Question Explanation</title> <style> body { font-family: Arial, sans-serif; margin: 20px; line-height: 1.6; } .formula { background-color: #f4f4f4; padding: 10px; border-radius: 5px; margin-top: 20px; } </style> </head> <body> <h1>Explanation for MCQ Involving a Right-Angled Triangle</h1>

The given question is:

A right-angled triangle has a hypotenuse of length 10 cm, and one of its acute angles measures $30^\circ$. What are the lengths of the other two sides?

Let's solve this step-by-step.

<h2>Given Information:</h2>

Hypotenuse ($c$) = 10 cm
One acute angle = $30^\circ$

<h2>Right-Angled Triangle Properties:</h2>

In a right-angled triangle, the relationship between the sides and angles is governed by trigonometric ratios. For a triangle with one angle measuring $30^\circ$, we can deduce the following using known properties of $30^\circ-60^\circ-90^\circ$ triangles:

The side opposite the $30^\circ$ angle (shorter leg) is half the hypotenuse.
The side opposite the $60^\circ$ angle (longer leg) is $\sqrt{3}$ times the shorter leg.

<h2>Calculation:</h2>

Let's denote the side opposite the $30^\circ$ angle as $a$ and the side opposite the $60^\circ$ angle as $b$. Given the properties mentioned earlier, we have:

Substituting the given hypotenuse $c = 10 \text{ cm}$, we get:

<div class="formula"> \[ a = \frac{10}{2} = 5 \text{ cm} \] \[ b = 5 \cdot \sqrt{3} = 5\sqrt{3} \text{ cm} \] </div> <h2>Conclusion:</h2>

The lengths of the other two sides, considering one angle measures $30^\circ$, are:

$a = 5 \text{ cm}$
$b = 5\sqrt{3} \text{ cm}$

Therefore, the correct choice among the given options is:

$5 \text{ cm and } 5\sqrt{3} \text{ cm}$.

</body> </html>

A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles measures
$30^\circ$
.
What are the lengths of the other two sides?

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