Final answer:
The question is about proving the Law of Sines using the altitudes in a triangle. After establishing the right triangles formed by the altitudes, one needs to set up ratios of the side length to the sine of the opposite angle and demonstrate that they are equal. This effectively proves the Law of Sines by illustrating the constant ratio across all sides and angles.
Step-by-step explanation:
The question involves proving the Law of Sines using the properties of a triangle with altitudes. Given the lengths of the sides (xy, xz, yz) as z, y, x, respectively, and altitudes (h1 and h2), Eric and Maggie need to relate these lengths to the angles of the triangle using trigonometric functions such as sine, cosine, and tangent as described by their respective ratios in a right triangle.
To progress with the proof after the altitudes are drawn, we consider each right triangle formed by the altitudes separately. In the right triangle with the hypotenuse yz (x) and altitude h1, sine of angle Y is h1 divided by the hypotenuse x. Similarly, in the triangle with hypotenuse xz (y) and altitude h2, sine of angle Z is h2 divided by hypotenuse y. These expressions can be rearranged to illustrate the Law of Sines, which states that the ratio of the length of a side to the sine of the opposite angle is constant for all sides and angles in a given triangle.
To complete the proof, we would set up the ratios of each side length to the sine of its respective opposite angle and show that these ratios are equal, thus proving the Law of Sines. This involves deriving sine Y = h1 / x and sine Z = h2 / y from the mentioned right triangles and comparing them with sine X using triangle XYZ to establish the law: (sin X / x) = (sin Y / y) = (sin Z / z).