A mason is building a semielliptical fireplace arch that has...

A mason is building a semielliptical fireplace arch that has a height of 2 feet at the center and a width of 6 feet along the base (see figure). The mason draws the semiellipse on the wall by the method shown on page $708 .$ Find the positions of the thumbtacks and the length of the string.

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Key Concepts

String Construction Method

The string method, also known as the gardener’s method, is a practical way to draw an ellipse. It involves placing two pins at the foci, looping a string around them, and then using a pencil to trace the curve while keeping the string taut. The total length of the string must equal the length of the major axis of the ellipse. This technique clearly illustrates the defining property of an ellipse where the sum of the distances from any point on the ellipse to the foci remains constant.

Ellipse Geometry

Ellipses are defined as the set of points for which the sum of the distances to two fixed points, called the foci, is constant. This constant is equal to the length of the major axis of the ellipse (2a), and the relationship between the semi-major axis (a), semi-minor axis (b), and the focal distance (c) is given by c² = a² - b². This concept is fundamental to understanding the shape and dimensions of ellipses in geometry.

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