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 longer leg is 24 yards
Find the length of the hypotenuse and the length of the shorter
leg. Give the exact answer and then an approximation to two
decimal places.

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Key Concepts

Exact Versus Approximate Calculations

When solving geometric problems, it is often important to distinguish between exact and approximate values. Exact answers, typically expressed in terms of radicals or other algebraic expressions, preserve mathematical accuracy, while approximate values, such as decimal approximations rounded to a specified number of places, provide a practical estimate or insight into the magnitude of the measurements.

30-60-90 Triangle

A 30-60-90 triangle is a type of special right triangle where the angles are exactly 30 degrees, 60 degrees, and 90 degrees. This triangle has well-defined side relationships that are derived from its angle measures, making it easier to solve problems that involve finding missing side lengths without resorting to trigonometric functions.

Side Ratios in Special Right Triangles

The sides of a 30-60-90 triangle are proportional to 1 : ?3 : 2, where the smallest side (opposite the 30° angle) is considered as the base unit. The longer leg, which is opposite the 60° angle, is ?3 times the shortest leg, and the hypotenuse (opposite the 90° angle) is twice the shortest leg. These ratios are essential in determining unknown side lengths when one side is known.

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