NRICH: Pythagoras Perimeters

If you know the perimeter of a right angled triangle, what can you say about the area?

If this right-angled triangle has a perimeter of 12 units, it is possible to show that the area is 366c 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: +2ab+=14424c+

b) So Area of the triangle =366c

c) a+b=12c

d) So 2ab=14424c

e) Area of the triangle =ab/2

f) By Pythagoras’ Theorem, +=

g) a+b+c=12

h) Dividing by 2ab=7212c
Printable Version

Can you adapt your method, or the method above, to prove that when the perimeter is 30 units, the area is 22515c square units?

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.

Age 14 to 16

Math Topics
Algebra & Pre-Algebra, Geometry, Quadratic Equations
High School, Educator
9th Grade, 10th Grade, 11th Grade
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