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Kissing Triangles

Determine the total shaded area of the 'kissing triangles'.

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Isosceles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

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Linkage

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Quadrilaterals in a Square

Age 11 to 14 Challenge Level:
Sam from Huntingdon School, Chenthuran from Chamblee Charter High School, Nasif from Wilson's School, Zach, Pablo from Kings College Alicante, Katie at Dulwich College Seoul, Aryan from Dumpton School and Matthew from Lochcarron Primary School all sent in some great work, and thought really hard about the problem. Thank you everybody!

Pablo from Kings College Alicante submitted a solution here, proving algebraically and geometrically the first and third parts of the question.

Thank you Pablo!

Here is another solution, from Katie at Dulwich College Seoul, proving all three parts of the question.

Thank you Katie!

Here is a solution from Rubaiyat from Wilson's School, which addresses each part of the question in a very clear way.

Thank you Rubaiyat!