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**tony123****Member**- Registered: 2007-08-03
- Posts: 225

Prove that if two right-angled triangles

have the same perimeter and the same area,

then they are congruent.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,648

hi tony123

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

Bob

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

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**tony123****Member**- Registered: 2007-08-03
- Posts: 225

nice work

thanks Bob

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**Hicies87****Member**- From: Portsmouth
- Registered: 2022-01-22
- Posts: 15

Great solution, thanks!

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**pamshaw****Member**- Registered: 2021-12-07
- Posts: 21

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.

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