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The acute angles of a right triangle have a ratio of $\frac{12}{3} .$ What is the difference between the two angle measures?

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42^{\circ}

54^{\circ}

64^{\circ}

72^{\circ}

answer is B.

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Detailed Solution

Given, the ratio of acute angles of a right-angled triangle is

\frac{12}{3}

.
Let the measure of acute angles be

12 x

and

3 x

respectively.
By angle sum property of a triangle,

12 x + 3 x + 90 = 180

\Rightarrow 15 x + 90^{\circ} = 180^{\circ}

\Rightarrow 15 x = 90^{\circ}

\Rightarrow x = 6^{\circ}

\Rightarrow 12 x = 12 \times 6^{\circ}

= 72^{\circ}

\Rightarrow 3 x = 3 \times 6^{\circ}

= 18^{\circ}

Therefore, the acute angles are

72^{\circ}

and

18^{\circ}

.
Difference between the angles is

72^{\circ} - 18^{\circ} = 54^{\circ} .

Hence, the correct option is 2.

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