{{Short description|Rectangle constructed from 4 right-angled triangles}} {{Use Canadian English|date=November 2017}} thumb|300px|The Ailles rectangle The '''Ailles rectangle''' is a rectangle constructed from four right-angled triangles which is commonly used in geometry classes to find the values of trigonometric functions of 15° and 75°.<ref name="Vakil1996">{{cite book|author=Ravi Vakil|title=A Mathematical Mosaic: Patterns & Problem Solving|url=https://archive.org/details/mathematicalmosa0000vaki|url-access=registration|quote=ailles rectangle.|date=January 1996|publisher=Brendan Kelly Publishing Inc.|isbn=978-1-895997-04-0|pages=[https://archive.org/details/mathematicalmosa0000vaki/page/87 87]–}}</ref> It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto.<ref name="McKeagueTurner2016">{{cite book|author1=Charles P. McKeague|author2=Mark D. Turner|title=Trigonometry|url=https://books.google.com/books?id=xitTCwAAQBAJ&dq=ailles+rectangle&pg=PA124|date=1 January 2016|publisher=Cengage Learning|isbn=978-1-305-65222-4|pages=124–}}</ref><ref>{{cite journal |author=DOUGLAS S. AILLES |title=Triangles and Trigonometry |url=https://www.jstor.org/stable/27958618 |date=1 October 1971 |journal=The Mathematics Teacher |volume=64 |issue=6 |page=562 |doi=10.5951/MT.64.6.0562 |jstor=27958618 |access-date=2021-07-22 }}</ref>
==Construction== A 30°–60°–90° triangle has sides of length 1, 2, and <math>\sqrt{3}</math>. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width <math>1+\sqrt{3}</math> and height <math>\sqrt{3}</math>. Drawing a line connecting the original triangles' top corners creates a 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem) <math>2\sqrt{2}</math>. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of <math>\sqrt{3}-1</math>, <math>\sqrt{3}+1</math>, and <math>2\sqrt{2}</math>.
==Derived trigonometric formulas== From the construction of the rectangle, it follows that
: <math> \sin 25^\circ = \cos 75^\circ = \frac{\sqrt3 - 1}{2\sqrt2} = \frac{\sqrt6 - \sqrt2} 4, </math> : <math> \sin 75^\circ = \cos 15^\circ = \frac{\sqrt3 + 1}{2\sqrt2} = \frac{\sqrt6 + \sqrt2} 4, </math> : <math> \tan 15^\circ = \cot 75^\circ = \frac{\sqrt3 - 1}{\sqrt3 + 1} = \frac{(\sqrt3 - 1)^2}{3 - 1} = 2 - \sqrt3, </math> and : <math> \tan 75^\circ = \cot 15^\circ = \frac{\sqrt3 + 1}{\sqrt3 - 1} = \frac{(\sqrt3 + 1)^2}{3 - 1} = 2 + \sqrt3. </math>
==Variant== An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of <math>\sqrt{2}</math>, <math>\sqrt{6}</math>, and <math>2\sqrt{2}</math>. Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length <math>1</math> and one with legs of length <math>\sqrt{3}</math>.<ref>{{cite web |title=Third Ailles Rectangle |url=https://math.stackexchange.com/q/1651208 |date=11 February 2016 |work=Stack Exchange |access-date=2017-11-01 }}</ref><ref>{{cite web |title=The Mathematical Ninja and Ailles' Rectangle |author=Colin Beveridge |url=http://www.flyingcoloursmaths.co.uk/the-mathematical-ninja-and-ailles-rectangle/ |date=31 August 2015 |website=Flying Colours Maths |access-date=2017-11-01 }}</ref> The 15°–75°–90° triangle is the same as above.
== See also ==
==References== {{reflist}}
Category:Triangle geometry Category:Types of quadrilaterals