You may also like

problem icon

2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

problem icon

Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

problem icon

Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Pythagoras Perimeters

Age 14 to 16 Challenge Level:



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.