In a 30^∘-60^∘-90^∘ right triangle, the length of the...

In a $30^{\circ}-60^{\circ}-90^{\circ}$ right triangle, the length of the hypotenuse is
1.5 feet. To the nearest hundredth, find the length of the shorter
leg and the length of the longer leg. Give the exact answer and
then an approximation to two decimal places, when
appropriate.

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Key Concepts

Using Ratios to Determine Side Lengths

The side length ratios in a 30-60-90 triangle (1 : ?3 : 2) provide a proportional reasoning framework to relate all sides. When the hypotenuse or any other side is given, one can set up a proportion to find the remaining side lengths exactly by applying the known multiples, making it a powerful method in geometry problems.

Exact Values Versus Decimal Approximations

In many mathematical problems, providing an exact answer using radicals or fractions is preferred for precision, while a decimal approximation is useful for practical interpretation. Understanding how to transition between exact expressions and their numerical approximations is essential, especially when instructions specify a certain degree of precision such as to the nearest hundredth.

30-60-90 Triangle Properties

A 30-60-90 triangle is a special type of right triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees. This triangle has fixed side ratios where the side opposite 30° is the shortest, the side opposite 60° is ?3 times the shortest side, and the hypotenuse (side opposite 90°) is twice the shortest side. These relationships allow for quick calculations of unknown side lengths when one side is known.

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