Pythagoras Perimeters

If you know the perimeter of a right angled triangle, what can you say about the area?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Pythagoras Perimeters printable worksheet - proof sorting

Pythagoras Perimeters printable worksheet - whole problem


Image
Pythagoras Perimeters


If this right-angled triangle has a perimeter of $12$ units, it is possible to show that the area is $36-6c$ square units.

Can you find a way to prove it?

Once you've had a chance to think about it, click below to see a possible way to solve the problem, where the steps have been muddled up.

Can you put them in the correct order?

 

 



a) Squaring both sides: $a^2+2ab+b^2 = 144-24c+c^2$

b) So Area of the triangle $=36-6c$

c) $a+b=12-c$

d) So $2ab=144-24c$

e) Area of the triangle $= \frac{ab}{2}$

f) By Pythagoras' Theorem, $a^2+b^2=c^2$

g) $a+b+c=12$

h) Dividing by $2$: $ab=72-12c$

Printable Version

 



Can you adapt your method, or the method above, to prove that when the perimeter is $30$ units, the area is $225 - 15c$ square units?

Extension 

Can you find a general expression for the area of a right angled triangle with hypotenuse $c$ and perimeter $p$?

With thanks to Don Steward, whose ideas formed the basis of this problem.