Question

constructing a wonder of the world!

the alhambra, a palace in granada, spain, is revered for its harmonious design. the reason for this is that architects who built it had knowledge of complex geometry. the palace is built following specific ratios. an example how they used ratios to bring harmony to architecture is shown below.

1. the architects wanted to build the face of the palace with the height of the arches, marked at point \\(d\\), being half the height of the palace. they also wanted the entryway to be exactly in the middle of the palace, marked at point \\(f\\). the main architect chose the measurements for \\(\delta def\\). find the building measurements, in meters, for \\(\delta aei\\) to satisfy the architects design criteria. (6 points)

\\(de = 4\text{m}\\)
\\(ef = 3\text{m}\\)
\\(df = 5\text{m}\\)

\\(ae = \underline{\quad\quad}\\)
\\(ei = \underline{\quad\quad}\\)
\\(ai = \underline{\quad\quad}\\)

Explanation:

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<reasoning>

Calculate AE using the height ratio

\[

$$\begin{aligned} &DE = 4\text{ m} \\ &AE = 2 \times DE = 2 \times 4\text{ m} = 8\text{ m} \end{aligned}$$

\]

Calculate EI using the midpoint ratio

\[

$$\begin{aligned} &EF = 3\text{ m} \\ &EI = 2 \times EF = 2 \times 3\text{ m} = 6\text{ m} \end{aligned}$$

\]

Calculate AI using the Pythagorean theorem

\[

$$\begin{aligned} &AI = \sqrt{AE^2 + EI^2} \\ &AI = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\text{ m} \end{aligned}$$

\]
</reasoning>

<answer>

1. The architects wanted to build the face of the palace with the height of the arches, marked at point D, being half the height of the palace. They also wanted the entryway to be exactly in the middle of the palace, marked at point F. The main architect chose the measurements for \(\Delta DEF\). Find the building measurements, in meters, for \(\Delta AEI\) to satisfy the architects' design criteria.

\(DE = 4\text{m}\)
\(EF = 3\text{m}\)
\(DF = 5\text{m}\)

\(AE =\) <blank>\(8\text{ m}\)</blank>
\(EI =\) <blank>\(6\text{ m}\)</blank>
\(AI =\) <blank>\(10\text{ m}\)</blank>
</answer>

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Answer:

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<reasoning>

Calculate AE using the height ratio

\[

$$\begin{aligned} &DE = 4\text{ m} \\ &AE = 2 \times DE = 2 \times 4\text{ m} = 8\text{ m} \end{aligned}$$

\]

Calculate EI using the midpoint ratio

\[

$$\begin{aligned} &EF = 3\text{ m} \\ &EI = 2 \times EF = 2 \times 3\text{ m} = 6\text{ m} \end{aligned}$$

\]

Calculate AI using the Pythagorean theorem

\[

$$\begin{aligned} &AI = \sqrt{AE^2 + EI^2} \\ &AI = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\text{ m} \end{aligned}$$

\]
</reasoning>

<answer>

1. The architects wanted to build the face of the palace with the height of the arches, marked at point D, being half the height of the palace. They also wanted the entryway to be exactly in the middle of the palace, marked at point F. The main architect chose the measurements for \(\Delta DEF\). Find the building measurements, in meters, for \(\Delta AEI\) to satisfy the architects' design criteria.

\(DE = 4\text{m}\)
\(EF = 3\text{m}\)
\(DF = 5\text{m}\)

\(AE =\) <blank>\(8\text{ m}\)</blank>
\(EI =\) <blank>\(6\text{ m}\)</blank>
\(AI =\) <blank>\(10\text{ m}\)</blank>
</answer>

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