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# Pythagorean Theorem.

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An isosceles right triangle is removed from each corner of a square piece of paper, as shown, to create a rectangle. If \(AB=12\) units, what is the combined area of the four removed triangles, in square units?

Jul 6, 2022

#1
-1

The combined area of the four removed triangles is 60.

Jul 6, 2022
#2
+7

nerdiest  Jul 6, 2022
#3
+7

image not shown.

look below

nerdiest  Jul 6, 2022
edited by nerdiest  Jul 6, 2022
#4
+1

Nothing shown......   Jul 6, 2022
#6
+8 here. SORRY. no words to explain. how dumb. i am.

nerdiest  Jul 6, 2022
edited by nerdiest  Jul 6, 2022
#5
+2

An isosceles right triangle is removed from each corner of a square piece of paper, as shown, to create a rectangle.

"as shown" huh?  There's nothing shown.

As long as this is a guessing game, my submission is either 36 or 144.  Pick one.

Jul 6, 2022
#7
+8

my bad. you can downvote me. no words to explain. how dumb. i am.

nerdiest  Jul 6, 2022
#13
+2

nerdi, since I'm posting as a guest, I'm not allowed to vote.  I wouldn't down vote you anyway.

.

Guest Jul 7, 2022
#14
+8

thx and sorry

nerdiest  Jul 7, 2022
#8
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Call the equal sides of  the  small isosceles triangle  ,  a

Call the equal sides of the larger isosceles triangle, b

The short side of the  rectangle in the middle =  sqrt (a^2 + a^2)  =  sqrt (2a^2) = a sqrt (2)

The long side of  the rectangle in the  middle = sqrt (b^2 + b^2)  =sqrt (2b^2)  = b sqrt (2)

By the Pythagorem Theorem

[ asqrt (2)]^2  +   [b sqrt (2) ]^2  = AB^2

[a sqrt (2) ] ^2  + [ b sqrt (2) ] ^2  = 12^2

2a^2 + 2b^2  =  144    divide by 2

a^2 + b^2  = 72

But a^2 = the area of the two smaller isosceles triangles

And b^2 = the area of the larger two isosceles triangles

So....their combined area =    72   Jul 6, 2022
edited by CPhill  Jul 6, 2022
edited by CPhill  Jul 6, 2022
#9
+9

.......

no words to explain how genius you are.

nerdiest  Jul 6, 2022
#10
+3

LOL!!!

Thx....I just get lucky once in a while  !!!!!   CPhill  Jul 6, 2022
#11
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That's a really neat solution, Chris!

BuilderBoi  Jul 6, 2022
#12
+3

Thx, BuilderBoi  !!!!!   CPhill  Jul 6, 2022