Question

a building manager needs to order 9 replacement panes that are all the same size, each similar to the window itself. at what angles should each pane be cut in order to fit in the window? what are the dimensions of each pane? explain.
the angle measures for each pane are 42,42,96. (type integers or decimals. do not include the degree symbol in your answers. use a comma to separate answers as needed.)
the side lengths of each pane are 13.5 ft. the angles are the same by the definition of similarity. the sides are one - third the length of the entire window because four identical panes fit along each side. (type integers or decimals. use a comma to separate answers as needed.)

Explanation:

Step1: Analyze the angle measures

Since the panes are similar to the window (by the definition of similarity, corresponding angles of similar triangles are equal). The window has angles \(42^{\circ}, 42^{\circ}, 96^{\circ}\) (using the angle - sum property of a triangle \(180-(42 + 42)=96\)). So the angle measures of each pane are \(42,42,96\).

Step2: Analyze the side - length relationship

The side lengths of the window are \(13.5\) ft (assuming one of the sides). If there are \(n\) identical panes. Let's assume the side - length of the pane is \(x\) and of the window is \(L\). Given that the sides of the pane are one - third of the window. If the window is divided into \(n\) panes such that the side - length relationship holds. Since the ratio of side - lengths of similar triangles (pane and window) is \(1:3\), and if we consider the number of panes along each side. If we assume a triangular window divided into smaller similar triangles. Using the property of similar triangles and area or side - length ratios. If the side - length ratio is \(k = \frac{1}{3}\), and if we consider the number of non - overlapping similar sub - triangles (panes) that fit into the larger triangle (window). The formula for the number of non - overlapping similar sub - triangles with side - length ratio \(k\) in a larger similar triangle is \(n=\frac{1}{k^{2}}\) (from the area ratio, since area \(A\propto s^{2}\), where \(s\) is the side - length). Here \(k=\frac{1}{3}\), so \(n = 9\) (since \(9\) panes are ordered). The number of panes along each side is \(3\) (because \(3\times3=9\) for a triangular arrangement of similar sub - triangles).

Answer:

The angle measures are \(42,42,96\). The side lengths of each pane: if the window has a side - length of \(13.5\) ft, then the side - length of each pane is \(\frac{13.5}{3}=4.5\) ft. The panes have angles \(42^{\circ},42^{\circ},96^{\circ}\) (because of the similarity of the panes to the window, corresponding angles of similar triangles are equal) and side - lengths that are one - third of the window's side - lengths (based on the number of panes (\(9\)) and the similarity ratio).