Determine the total shaded area of the 'kissing triangles'.
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?
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?
Several students from West Flegg worked on
this one by drawing . We apologise to those who were annoyed
because the number 10 did not seem to come into the solution.
The triangle inequality states: "the length of
one side of a triangle is always less than the sum of the lengths
of the other two sides of the triangle". Draw a diagram and
convince yourself that this has to be true.
So, using the triangle inequality, AB
< AP + PB = 3 + 4 = 7 .
As triangle ABC is equilateral we also
know now that CA < 7 and CB < 7. Now applying the triangle
inequality to triangle CAP, and using the fact that CA < 7,
gives CP < CA + AP < 7 + 3 = 10.
This shows that CP is less than 10 cm but it
is obviously quite a lot less than 10 cm.
To get a better estimate, draw a circle,
centre C, radius 7 centimetres, then since CA and CB are both less
than 7 centimetres the whole of triangle ABC is inside the circle.
As point P is inside triangle ABC it must be inside the circle and
hence CP < 7 cm.