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# Prove this Geometry Problem

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3 Jun 5, 2020

#1
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By the Pythagorean Theorem, we know that  x2 + h2  =  a2   and   y2 + h2  =  b2

which, if we add them together, we get:    x2 + h2  +  y2 + h2  =  a2 + b2

However, we want to show that (x + h)2 + (y + h)2  =  (a + b)2

and, this is:      ( x2 + 2xh + h2 ) + ( y2 + 2yh + h2 ) =  a2 + 2ab + b2

So, somehow we will need to show tha 2xh + 2yh  =  2ab

We can do this by finding the area of the triangle in two different ways:

Area  =  ½ab   and the area is also   Area  =  ½(x +y)h

So:  ½(x +y)h  =  ½ab

--->     (x + y)h  =  ab

--->     xh + yh  =  ab

--->   2xh + 2yh  =  2ab

Adding this to:  x2           + h2  +  y2           + h2  =  a2          + b2

we get:             x2 + 2xh + h2  +  y2 + 2yh + h2  =  a2 + 2ab + b2

which is:              (x + h)2        +     (y + h)2        =      (a + b)2

Jun 5, 2020
#3
+1

thank you so much geno3141

NoWorkNeeded  Jun 5, 2020
#2
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Let's use real numbers.

a = 3      b = 4      c = 5      h = 2.4       x = 1.8       y = 3.2

(x + h)² + (y + h)² = (a + b)²

(1.8+2.4) + (3.2+2.4)  = (3 + 4)²

17.64 + 31.36 = 49

49 = 49 Jun 5, 2020