Rectangle ACEF shown in the figure is a golden rectangle. It is constructed from square ACDB by holding line segment OB fixed at point O and then letting point B drop down until OB aligns with CD. The ratio of the length to the width in the golden rectangle is called the golden ratio. Find the lengths below to arrive at the golden ratio. (Let a = 6.) ## Exploring the Golden Ratio in Geometry### IntroductionThe rectangle \( ACEF \) shown in the figure below is a golden rectangle. It is constructed from the square \( ACDB \) by holding line segment \( OB \) fixed at point \( O \) and then letting point \( B \) drop down until \( OB \) aligns with \( CD \). The ratio of the length to the width in the golden rectangle is called the golden ratio. In this exercise, we will find the lengths of specific segments to demonstrate the golden ratio. Assume \( a = 6 \) units.### Diagram ExplanationIn the diagram:- \( ACDB \) is a square with side length \( a \).- \( O \) is the midpoint of \( CD \).- \( OB \) is a diagonal of the square.- \( B \) drops down perpendicularly to meet \( E \) on \( CD \).- The distance \( OD = OC = \frac{a}{2} \).- The rectangle \( ACEF \) is formed with \( CE \) extended to match \( EB \).### Questions and Calculations#### (a) Find the length of \( OB \).\[ OB = \sqrt{a^2 + \left(\frac{a}{2}\right)^2} = \sqrt{a^2 + \frac{a^2}{4}} = \sqrt{\frac{5a^2}{4}} = \frac{a\sqrt{5}}{2} \]Given \( a = 6 \):\[ OB = \frac{6 \sqrt{5}}{2} = 3\sqrt{5} \]#### (b) Find the length of \( OE \).\[ OE = OB = 3\sqrt{5} \]#### (c) Find the length of \( CE \).The length \( CE \) is \( AC + AE \):\[ CE = a + 2a = 3a \]Given \( a = 6 \):\[ CE = 3 \times 6 = 18 \]#### (d) Find the ratio \( CE/EF \).Since \( \text{EF} = a = 6 \), the ratio is:\[ \frac{CE}{EF} = \frac{18}{6} = 3 \]Thus, we have verified the elements contributing to the golden ratio

Name: Rectangle ACEF shown in the figure is a golden rectangle. It is constr
Uploaded: 2024-03-20T06:47:08.000Z
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Description: Rectangle ACEF shown in the figure is a golden rectangle. It is constructed from square ACDB by holding line segment OB fixed at point O and then letting point B drop down until OB aligns with CD. The ratio of the length to the width in the golden re