GEOMETRIA PRACTICE WITH CLIL Golden Rectangle The golden rectangle is a rectangle whose side lengths are in the golden ratio. Let us see how to construct one. 4. From point E, draw a line perpendicular to AB that intersects the line CD in F. D C F B E 1. Draw an ABCD square of side l. D C A A B M 5. Now prove that the rectangle AEFD is a golden 2. Draw the axis of symmetry that shows point M of side AB. rectangle, that is its side lengths AD and AE are in the golden ratio. D D C F B E C A A M B 6. Apply the Pythagoras Theorem to the right-angled 3. Rotate segment MC, with extreme in M, until it lines up with the extension of AB, on the B side, and draw point E. D M C triangle MBC and determine its hypotenuse MC. ________ l 2 l _ MC = l2 + (__) = __ 5 2 2 MC ME to construct _ l l _ l AE = __ + __ 5 = __(1 + 5) 2 2 2 The relation between the lengths of the rectangle AEFD: 2 _ l AD _________ ___ _______ = _ = AE __l 1 + 5 1 + 5 A M B E ) 2( By rationalizing the denominator, you will get: _ _ 2( 5 1) 2( 5 1) _ 5 1 _______________ _ _ = _________ = _______ 4 2 (1 + 5)( 5 1) _ AD 5 1 Hence ___ = _______, that is the golden ratio. AE Exercise Show that the rectangle BEFC above is a golden rectangle. 354 2