This problem solving activity has a geometry focus.
Poppy is trying to find a method that might enable her to find the position of her helicopter inside a rectangular region.
She first experimented by drawing a 10cm by 13cm rectangle and measured a, b, c and d for various points P inside the rectangle.
She started the table below.
Complete the table and try to find a relationship between a, b, c and d.
Can you justify this relationship?
This problem has at least three aspects that are worth thinking about. First it gives the students a chance to construct and measure certain geometrical objects. Second it enables them to find a pattern in a geometrical situation. Third it gives them practice in constructing proofs using a well known result. The extension to this problem also provides the opportunity to look for a generalisation of the problem.
Looking for patterns is a key part of mathematics and mathematicians look for them in all possible situations. Knowing that this is important and practising this skill will provide a good basis for all future mathematics.
Note: The software programmes Geometer's Sketchpad or Cabri Geometry, will enable students to generate a lot of examples quickly. This will be less tedious (and more accurate) than choosing the point P and measuring a, b, c and d.
If you don't have access to either of these, the generation of the table can be accomplished more efficiently by sharing group measurements.
Poppy is trying to find a method that might enable her to find the position of her helicopter inside a rectangular region.
She first experimented by drawing a 10cm by 13 cm rectangle and measured a, b, c and d for various points P inside the rectangle. She started the table below.
Complete the table and try to find a relationship between a, b, c and d.
Can you justify this relationship?
What if Poppy’s helicopter is outside of the rectangular boundary? Does the relationship above still hold?
The relationship can be established by using Pythagoras’ Theorem. From the diagram above
a2 = u2 + v2 c2 = u2 + w2
b2 = x2 + w2 d2 = v2 + x2
So, a2 + b2 = (u2 + v2) + (x2 + w2)
= u2 + x2 + v2 + w2
= c2 + d2.
Surprisingly, the same relations holds.
Again, by Pythagoras,
a2 = (x + z)2 + v2, b2 = w2 + z2
c2 = w2 + (x + z)2and d2 = y2 + z2
a2 + b2 = (x + z)2 + y2 + w2 + z2
= w2 + (x + z)2 + y2 + z2
= c2 + d2.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/poppy-meets-pythagoras at 8:58pm on the 26th February 2024