This problem solving activity has a geometry focus.
Julie stacks three different sized wheels against the shed.
Each wheel fits together neatly, like this.
The radius of the largest wheel is 16cm and the radius of the middle-sized wheel is 9cm.
What is the radius of the smallest wheel?
This problem challenges students to work out how to apply Pythagoras’ Theorem to a situation in which the centres of three circles do not lie on a right-angled triangle. Finding a solution involves students in drawing a diagram, connecting points, ascribing letters to points, and applying algebra.
Julie stacks three different sized wheels against the shed.
Each wheel fits together neatly, like this.
The radius of the largest wheel is 16cm and the radius of the middle-sized wheel is 9cm.
What is the radius of the smallest wheel?
Can you find a formula for the radius of the smallest wheel in terms of the radii of the other two wheels?
In other words, if you are given the radii of the two larger circles as a and b, can you find c in terms of a and b?
To apply Pythagoras, the triangles must be found. Let the radius of the smaller wheel be c. We now need three equations.
In triangle ABB' , AB = 16 + 9 = 25 and AB' = AD – BC = 16 – 9 = 7,
(BB' )2 = AB 2 – (AB' )2 = 252 – 72 = 576.
In triangle AEF' , AE = 16 + c and AF' = 16 – c, and
(F' E)2 = AE2 – (AF' )2 = (16 + c)2 – (16 – c)2 = 64c,
In triangle BEF, BE = 9 + c and BF = 9 – c, and
FE2 = BE2 – BF2 = (9 + c)2 – (9 – c)2 = 36c.
But BB' = F'F + EF, so
√576 = √64c + √36c
√576 = 8√c + 6<√c
14√c = 24
... √c = 12/7 ... √c = 144/49.
c = ab/(<√a + <√b)2.
Printed from https://nzmaths.co.nz/resource/julie-s-wheels at 2:40pm on the 27th April 2024