In a 30^∘-60^∘-90^∘ triangle, the length of the shorter leg is 5 √(2) inches. Find the length of

step 1 Okay, so let's say we have a 30, 60, 90 triangle, and we know that the length of our shorter leg

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

In a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, the length of the shorter leg is
$5 \sqrt{2}$ inches. Find the length of the hypotenuse and the length
of the longer leg. Give the exact answer and then an
approximation to two decimal places.

Key Concepts

Operations with Radical Expressions

Working with radical expressions involves performing arithmetic operations while keeping expressions in a simplified, exact form using square roots and other radicals. This is essential in geometry and algebra to maintain precision, and later, these expressions can be approximated numerically if required.

Exact Versus Approximate Values

In many mathematical contexts, answers may be required in exact form or as decimal approximations. Exact values are preferred when precision is necessary, while approximate values, rounded to a specified degree of accuracy, are useful for practical applications and when a numerical estimate is needed.

Side Ratios in a 30-60-90 Triangle

A 30-60-90 triangle is a special right triangle with angle measures of 30°, 60°, and 90°. The sides of this triangle are in a fixed ratio: the side opposite the 30° angle (the shorter leg) is taken as 'x', the side opposite the 60° angle (the longer leg) is 'x?3', and the hypotenuse is '2x'. This property allows one to determine all side lengths if at least one is known.

Special Right Triangles

Special right triangles are triangles with angles and side ratios that are fixed and well-known. These triangles allow for quick calculations without resorting to the Pythagorean theorem directly, making them a powerful tool in geometry. Their predictable ratios simplify solving problems involving trigonometry and geometric constructions.

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