Two shaded identical rectangular decorative tiles are first placed (one each) at the top and at the base of a door frame for a hobbit's house, as shown in Figure 1. The distance from W to H is 45 inches. Then the same two tiles are rearranged at the top and at the base of the door frame, as shown in Figure 2. The distance from Y to Z is 37 inches. What is the height of the door frame, in inches?
(x + 1)^2 + (y - 1)^2 = (8.5)^2
To find the height of the door frame, we need to determine the height of each rectangular tile.
Let's label the top-left corner of Figure 1 as point A, the top-right corner as point B, the bottom-left corner as point C, and the bottom-right corner as point D. We'll also label the top-middle point of Figure 1 as point E and the bottom-middle point as point F.
Similarly, in Figure 2, let's label the top-left corner as point G, the top-right corner as point H, the bottom-left corner as point I, and the bottom-right corner as point J. We'll label the top-middle point of Figure 2 as point K and the bottom-middle point as point L.
As per the problem statement, we're given that the distance from point W to point H is 45 inches and the distance from point Y to point Z is 37 inches.
Using the information from Figure 1, we know that the distance between points A and B is the width of the rectangular tile, denoted as AB. Similarly, the distance between points C and D is also the width of the tile.
In Figure 2, the distance between points G and H is the width of the tile, denoted as GH. Similarly, the distance between points I and J is also the width of the tile.
To find the height of the door frame, we need to determine the distance between points C and G in Figure 1 and the distance between points I and A in Figure 2. Let's denote the height as h.
Using the information given, we can set up the following equations:
AC + CG = 45 (from Figure 1)
IA + AG = 37 (from Figure 2)
Since the width of the tile stays the same, we can say that:
AB = GH
CD = IJ
Now, if we look at Figure 1, we can observe that the height of the tile, denoted by h, can be represented as:
h = EF
Similarly, in Figure 2, the height of the tile, denoted by h, can be represented as:
h = KL
Now, using Pythagoras' theorem, we can determine the values of AB, CD, IA, and HC:
(AC^2) + (CG^2) = (AG^2) (from right triangle ABC)
(IA^2) + (AG^2) = (HG^2) (from right triangle IGH)
Since AB = GH and CD = IJ:
(AC^2) + (CG^2) = (IA^2) + (HC^2)
Now we have a system of equations:
AC + CG = 45
IA + AG = 37
(AC^2) + (CG^2) = (IA^2) + (HC^2)
By solving these equations simultaneously, we can determine the value of h, which will give us the height of the door frame.