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 \\(\square\\) ft. the angles are the same by the definition of \\(\square\\) the sides are \\(\square\\) the length of the entire window because identical panes fit along each side. (type integers or decimals. use a comma to separate answers as needed.)

Explanation:

Step1: Confirm similar triangle angles

Similar triangles have congruent corresponding angles. The original window has angles $42^\circ$, $42^\circ$, $96^\circ$, so each pane has the same angles.

Step2: Count panes per side

The window is divided into 9 congruent similar triangles, forming a 3-by-3 grid of small triangles along each side. So 3 panes fit along each side.

Step3: Calculate pane side lengths

Divide each window side length by 3:
Long side: $\frac{60}{3}=20$ ft
Equal sides: $\frac{40.5}{3}=13.5$ ft

Answer:

The angle measures for each pane are 42,42,96.
The side lengths of each pane are 13.5, 13.5, 20 ft. The angles are the same by the definition of similar triangles. The sides are $\frac{1}{3}$ the length of the entire window because 3 identical panes fit along each side.