Transcript TXProving the Pythagorean Theorem

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This sample is an excerpt from the presentation Proving the Pythagorean Theorem in
High School Geometry, which contains 50 interactive presentations
in total.2014
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The Pythagorean theorem
The Pythagorean theorem:
In a right triangle, the square of the length of the hypotenuse
is equal to the sum of the square of the lengths of the legs.
The area of the largest
square is c × c or c2.
c2
a2
a
c
b
The areas of the smaller
squares are a2 and b2.
The Pythagorean theorem
can be written as:
b2
c2 = a2 + b2
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Showing the Pythagorean theorem
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A proof of the Pythagorean theorem
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An altitude of a triangle
What is an altitude of a triangle?
Z
An altitude of a triangle is
a perpendicular line segment
from a side to the opposite vertex.
W
Y
X
All triangles have three altitudes.
This figure shows one altitude of
a right triangle.
The Pythagorean theorem can be proved using altitudes and
similar triangles.
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Proof using similarity (1)
Show that the three triangles in this figure are similar.
The altitude to the hypotenuse of a right triangle creates three
right triangles: △XYZ, △WXY and △WYZ.
∠XYZ and ∠XWY are
by definition:
Z
both right angles
by the reflexive property:
∠ZXY ≅ ∠WXY
by the AA
similarity postulate:
△XYZ ~ △WXY
by definition:

∠XYZ and ∠YWZ are
both right angles
Y
by the reflexive property:
∠XZY ≅ ∠WZY
by the AA
similarity postulate:
△XYZ ~ △WYZ

by the transitive
property of congruence:
△WXY ~ △WYZ

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Proof using similarity (2)
Show that a2 + b2 = (c + d)2 using similar triangles.
△XYZ ~ △YWZ ~ △XWY
Z
△XYZ is similar to △XWY:
d
b
=
b
c+d
c
b2 = d(c+d)
W
a
d
△XYZ is similar to △YWZ:
c
a
=
a
c+d
adding the two equations:
a2 = c(c+d)
Y
b
a2 + b2 = c(c+d) + d(c+d)
= c2 + cd + dc + d2
= c2 + 2cd + d2
= (c+d)2 
(c + d) is the hypotenuse of △XYZ, which proves
the Pythagorean theorem.
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