Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

congruent

Prove that if two right-angled triangles

have the same perimeter and the same area,
then they are congruent.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: congruent

hi tony123

Thanks for another interesting puzzle. I think this is a proof but it's a bit messy.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: congruent

nice work
thanks Bob

Hicies87

Member

From: Portsmouth

Registered: 2022-01-22

Posts: 15

Re: congruent

Great solution, thanks!

pamshaw

Member

Registered: 2021-12-07

Posts: 21

Re: congruent

The Question states that Prove that if two right-angled triangles: ABC, XYZ
have the same perimeter and the same area, then they are congruent.

Solution:
Assuming that AB = kXY
Assuming areas are equal, which means AB x BC = XY x YZ
AB = XY x YZ / BC
Putting AB = kXY
kBC = YZ
Let perimeter be 
P => CA = P - AB - BC
(P - AB - BC) ^2 = BC ^2 + AB ^2
P^2 = 2P.BC + 2P.AB + 2.AB.BC
Similarly, P^2 = 2P.YZ + 2P.XY + 2.XY.YZ
YZ + XY = AB + BC
This proves areas and perimeters of the two are equal
Put k=1 in which case, AB=XY and BC = YZ and the triangles are congruent
Note: Two segments are congruent if and only if they have equal measures. Two triangles are congruent if and only if all corresponding angles and sides are congruent.