Project 3.1 The Crescent
A crescent is made up of two circles, as shown in the diagram. O
is the centre of the larger circle and C is the centre of the
smaller circle.
1. If the width of the crescent between T and S
is 10 cm, and between P and Q is 6 cm, show that r
= R - 5 where r is the radius of the smaller circle and R
is the radius of the larger circle.
2. Determine the diameters of the two circles.
3. Create a crescent of your own and determine the
diameters of the two circles.
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Project 3.2 Three Circles
A circle with centre O and radius 12 cm is drawn, as shown in
the figure. A second circle with centre A and radius 6 cm is
drawn having its diameter as a radius of the first circle. A third
circle with centre B and radius r cm is now drawn as shown
to touch the first circle internally, the second circle externally and the
tangent which meets the second circle at O.  |
1. Express OB in terms of r.
2. Express AB in terms of r.
Let BM be perpendicular to OA so that OMBC is a
rectangle.
3. Express AM in terms of r.
6. What is the value of r?
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Project 3.3 Diophantine Problem
Pythagoras' Theorem states that in a right-angled triangle the square
on the hypotenuse is equal to the sum of the squares on the other two
sides. That is:
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One possible solution of this
equation is
1. Check that these values give a solution for the equation
above.
2. Find, by trial and error, two more solutions to this equation.
Equations that have integer solutions are known as Diophantine
equations, after the Greek mathematician, Diophantos. He lived
in the third century and invented modern algebra. |
Diophantos studied equations of the form
for some, n, and where x and y are integers.
E.g. when n = 8, we have
3a. If x and y are positive integers, show that
can be expressed as
3b. List the possible integral factors of 64.
3c. Find all positive integers x and y so that
by completing the following table:
3d. Check your answers by substituting the values
of x and y in the equation
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