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If in a right triangle, the ratio of the shortest side to the longest is \(\frac{1}{2}\), what is the smallest angle in this triangle?

A. 15 degrees B. 20 degrees C. 30 degrees D. 45 degrees E. 60 degrees

Since the ratio of hypotenuse to the smallest leg in a right triangle is \(\frac{1}{2}\), then it means that we have 30°-60°-90° right triangle. So, the smallest angle is 30°.

A right triangle where the angles are 30°, 60°, and 90°.

This is one of the 'standard' triangles you should be able to recognize on sight. A fact you should commit to memory is: The sides are always in the ratio \(1 : \sqrt{3}: 2\).

Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°).

Let the three sides be a, b & c. where a<b<c or a=b <c

& \(a^2 + b^2 = c^2\) since the given triangle is a right triangle.

Now as per the question : \(\frac{a}{c}\) = \(\frac{1}{2}\) Square both the sides: \(\frac{a^2}{c^2}\) = \(\frac{1}{4}\) putting the value of \(c^2\) : \(\frac{a^2}{a^2+b^2}\) = \(\frac{1}{4}\)