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Kissing Triangles

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

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Trice

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

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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?

Weekly Problem 13 - 2012

Stage: 3 Challenge Level: Challenge Level:1

$\angle BAC = 90^{\circ}$

$A$, $B$ and $C$ are all equidistant from $D$ and therefore lie on a circle whose centre is $D$. $BC$ is a diameter of the circle and $\angle BAC$ is therefore the angle subtended by a diameter at a point on the circumference (the angle in the semicircle).

(Alternatively: let $\angle ACD = x$ and show that $\angle DAC = x,\, \angle ADB = 2x$ and $\angle DAB = 1/2 (180^{\circ}) - 2x = 90^{\circ} - x$. Hence $\angle BAC = x + 90^{\circ} - x = 90^{\circ}$.)

This problem is taken from the UKMT Mathematical Challenges.

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