548 /www/nrich/html/content/98/01/six3/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><p>The triangle ABC is isosceles with a right angle at B. In each diagram the triangle ABC has either a circle or one of two (different) squares inscribed within it.</p> <p><mdo:image alt="" src="areas.jpg"></mdo:image></p> <p>Which of the inscribed figures has the greatest area?</p> <p><a href="http://nrich.maths.org/7998">Click here for a poster of this problem</a>.</p> <p> </p></mdoxml> <mdoxml version="1.0"><br></br> <p class="editorial">Many got the correct answer that the circle which is embedded inside the isosceles triangle is the largest of the three shapes.</p> <p class="editorial">Some students measured the diagrams given which luckily had been drawn accurately to scale, while others observed the symmetry of the isosceles triangle and the consequences of joining B to D the mid point of AC. To find the radius of the circle it helps to draw perpendiculars from the centre to the sides of the triangle.</p> <p class="editorial">Taking the lengths of the short sides of the triangle as $2$ units:</p> <p class="editorial">the radius of the circle is found to be $2-\sqrt{2}$ and the area to be $\pi (6 - 4\sqrt{2})$.</p> <p class="editorial">the side length of square one to be $1$ and area $1$;</p> <p class="editorial">and the side length of square two to be $\frac{2\sqrt{2}}{3}$ and area $\frac{8}{9}$.</p> <p><span class="editorial">Many students evaluated the ratio of the areas as $1.078 : 1: 0.889$ (to $3$ dec. places)</span></p> <p class="editorial"><span class="editorial">Be</span>n and Adrianfrom the Simon Langton Grammar School for Boys used trig ratios, as did Hannah of Maidstone Girls Grammar School, to arrive at the same conclusions.</p> <br></br></mdoxml> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=548" style="font-weight: bold;">this problem</a> <span style="font-weight: bold;">?</span></h3> This problem brings together ideas of areas of circles and squares, the use of Pythagoras theorem and the property of tangents to a circle from an exernal point.<br></br> <br></br> <h3><span style="font-weight: bold;">Possible approach</span></h3> You might start with the middle diagram which is the easiest. It brings in the ratio of the sides of an isosceles right angles triangle which is again used in the other two parts.<br></br> <br></br> <h3><span style="font-weight: bold;">Key questions</span></h3> If you know the side length of an isosceles right angled triange how do you find the hypotenuse?<br></br> <br></br> Which lengths are equal in the digram?<br></br> <br></br> Which angles are equal?<br></br> <br></br> Can you use the symmetry of the diagram?<br></br> <br></br> <h3><span style="font-weight: bold;">Possible extension</span></h3> The problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=2163&amp;part=index"> Circle-in</a> also uses one of the circle theorems (the tangent is perpendicular to the radius at the point of contact).<br></br></mdoxml> <mdoxml version="1.0">Draw in the radii of the circle. Use the symmetry in the diagrams.<br></br></mdoxml> 2 4 0 0 0 1 0 Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle? Measures and Mensuration Area 2D Geometry, Shape and Space Squares 2D Geometry, Shape and Space Circle theorems 2D Geometry, Shape and Space Pythagoras' theorem 2D Geometry, Shape and Space Inscribed polygons Information and Communications Technology smartphone