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Cow Modeling Problem found in some geometry texts:A cow is tied by a rope to the corner of a rectangular barn with dimensions __ by ___. Determine the maximum possible grazing area the cow has. Yet here, you can change the dimensions of the barn by dragging the corners of the rectangle.

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Cow Modeling Problem found in some geometry texts:A cow is tied by a rope to the corner of a rectangular barn with dimensions __ by ___. Determine the maximum possible grazing area the cow has. Yet here, you can change the dimensions of the barn by dragging the corners of the rectangle.

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In the diagram, six circles of equal size touch adjacent circles and the sides of the large rectangle.

Each of the corners of the small rectangle is the centre of one of the large circles.

The perimeter of the small rectangle is $60\text{cm}$.

What is the perimeter of the large rectangle?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

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In the diagram, six circles of equal size touch adjacent circles and the sides of the large rectangle. Each of the corners of the small rectangle is the centre of one of the large circles. The perimeter of the small rectangle is $60\text{cm}$. What is the perimeter of the large rectangle?

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ideas and resources for mathematics teachers of 10 to 16 year olds

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8 Basic Circle Theorems for Year 10 Advanced Maths. Mr. Wain

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Summary With the help of this activity, we have learnt that:  When A lies outside the circle then we can draw 2 tangents passing through A.  If A...

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8 Basic Circle Theorems for Year 10 Advanced Maths. Mr. Wain

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Proof without words: Median to hypotenuse

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Proof without words: Median to hypotenuse

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GHS 25  The Circle as a Whole and Its Parts

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https://www.geogebra.org/m/vvymwafu

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https://www.geogebra.org/m/bktkquv5

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Circle Theorems, Circle Properties, etc...

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The Crazy Circle Illusion: How are these dots moving? GeoGeb

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Investigate the circle theorems and corollaries.

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Some of the resources used in Ben's talk "Windmills of Your Mind" The Sunflower Spirals is also used in a Numberphile video on the Golden Ratio. Surd Spiral Mandelbrot Orbits Sunflower spirals

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Some of the resources used in Ben's talk "Windmills of Your Mind" The Sunflower Spirals is also used in a Numberphile video on the Golden Ratio. Surd Spiral Mandelbrot Orbits Sunflower spirals

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In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat-Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.

The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius , where and are the semi-major axis and semi-minor axis of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a right angle.[1]

The director circle of a hyperbola has radius , and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane.

More generally, for any collection of points , weights , and constant C, one can define a circle as the locus of points X such that

The director circle of an ellipse is a special case of this more general construction with two points P ₁ and P ₂ at the foci of the ellipse, weights w ₁ = w ₂ = 1, and C equal to the square of the major axis of the ellipse. The Apollonius circle, the locus of points X such that the ratio of distances of X to two foci P ₁ and P ₂ is a fixed constant r, is another special case, with w ₁ = 1, w ₂ = − r ², and C = 0.

In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola.

Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World, 26, American Mathematical Society, ISBN 978-0-8218-4323-9 .

Cremona, Luigi (1885), Elements of Projective Geometry, Oxford: Clarendon Press, p. 369.

Faulkner, T. Ewan (1952), Projective Geometry, Edinburgh and London: Oliver and Boyd

Hawkesworth, Alan S. (1905), "Some new ratios of conic curves", The American Mathematical Monthly, 12 (1): 1-8, doi:10.2307/2968867, MR 1516260.

Loney, Sidney Luxton (1897), The Elements of Coordinate Geometry, London: Macmillan and Company, Limited, p. 365.

Wentworth, George Albert (1886), Elements of Analytic Geometry, Ginn & Company, p. 150.

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In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat-Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other. The director circle of an ellipse circumscribes the minimum bounding box of the ellipse.

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Area of the circle by the triangle or the rectangle

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Area of the circle by the triangle or the rectangle

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This collection holds dynamic worksheets of all 8 circle theorems.

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Applets useful in exploring trigonometric unit circle symmetries

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Explore the formula for the area of a circle, by cutting it into sectors and rearranging the sectors to form a figure close to a parallelogram. By in...

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Circle definition vs. sphere definition: Interactive applet

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Tangente Interna Común Entre Dos Círculos - Proporción Entre Triángulos Semejantes

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Tangente Interna Común Entre Dos Círculos - Proporción Entre Triángulos Semejantes