Four Square. The game Four Square is making a comeback on...

Four Square. The game Four Square is making a comeback on college campuses. The game is played on a 16 -ft square court divided into four smaller squares that meet in the center. If a line is drawn diagonally from one corner to another corner, then a right triangle $Q T S$ is formed, where $\angle Q T S$ is $45^{\circ} .$ Using a trigonometric function, find the length of the diagonal. Round the answer to the nearest tenth of a foot.

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Key Concepts

Right Triangle

A right triangle contains one 90-degree angle, which allows for the use of trigonometric functions to relate its angles and side lengths. Recognizing a right triangle in a geometric figure is key to setting up equations to solve for unknown sides.

Trigonometric Ratios

Trigonometric functions like sine, cosine, and tangent express the relationships between the angles and sides of a right triangle. They are used to calculate unknown side lengths when specific angles and one or more side lengths are known.

Special Right Triangles (45-45-90 Triangles)

A 45-45-90 triangle is a special type of right triangle where the two acute angles are equal. This leads to a fixed ratio between the legs and the hypotenuse, with the hypotenuse being ?2 times the length of each leg—a property that simplifies calculations involving diagonals in squares.

Square Geometry and Diagonals

In a square, diagonals have unique properties: they bisect each other and the angles of the square, forming congruent right triangles. Understanding these geometric properties is essential when calculating the length of a diagonal, as it directly relates to the dimensions of the square.

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