Question : If the measures of the angles of a triangle are in the ratio 1 : 2 : 3, and if the length of the smallest side of the triangle is 10 cm, then the length of the longest side is:
Option 1: 20 cm
Option 2: 25 cm
Option 3: 30 cm
Option 4: 35 cm
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Correct Answer: 20 cm
Solution :
Given:
The measures of the angles of a triangle are in the ratio 1 : 2 : 3.
Let the angles be $x, 2x, 3x$.
So, $x + 2x + 3x = 180°$
⇒ $6x = 180°$
∴ $x = 30°$
So, the angles are 30°, 60°, and 90°.
Let $a, b,$ and $c$ units be the length of the sides opposite to the angles 30°, 60°, and 90° respectively.
Now applying the sine rule, we get,
$\frac{a}{\sin30°} = \frac{b}{\sin60°} = \frac{c}{\sin90°}$
⇒ $2a = \frac{2b}{\sqrt{3}}=c$
$2a=c$
⇒ $a=\frac{c}{2}$
Also, $\frac{2b}{\sqrt{3}}=c$
⇒ $b=\frac{\sqrt{3}c}{2}$
∴ $a:b:c=\frac{c}{2}:\frac{\sqrt{3}c}{2}:c=1:\sqrt{3}:2$
Smallest side = 1 unit = 10 cm
So, largest side = 2 units = 20 cm
Hence, the correct answer is 20 cm.
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