Answer:
Step-by-step explanation:By AA similarity postulate
△ADB∼△ABC∼△BDC
therefore the sides of the triangles are proportional, in particular
ADAB=ABAC ACBC=BCDC
By algebra we have the following equations
AD⋅AC=AB⋅ABAC⋅DC=BC⋅BC
this is the same as
AD⋅AC=AB2AC⋅DC=BC2
"Equals added to equals are equal" allows us to add the equations
AD⋅AC+AC⋅DC=AB2+BC2
By distributive property
AC(AD+DC)=AB2+BC2
but by construction AD+DC=AC.
Substituting we have
AC⋅AC=AB2+BC2
this is equivalent to
AB2+BC2=AC2
which is what we wanted to prove
The Pythagorean theorem uses similar triangles, This is equivalent to AB^2+BC^2=AC^2.
We have given that,
the converse of the Pythagorean theorem using similar triangles.
The converse of the Pythagorean theorem states that when the sum of the squares of the links of the legs of the triangle equals the shared length of the hypotenuse, the triangle is a right triangle.
[tex]hypotenuse ^2=side^2+side^2[/tex]
By AA similarity postulate
△ADB∼△ABC∼△BDC
Therefore the sides of the triangles are proportional, in particular
ADAB=ABAC ACBC=BCDC
By algebra, we have the following equations
AD⋅AC=AB⋅ABAC⋅DC=BC⋅BC
This is the same as
AD⋅AC=AB^2AC⋅DC=BC^2
Equals added to equals are equal allows us to add the equations
AD⋅AC+AC⋅DC=AB^2+BC^2
By distributive property
AC(AD+DC)=AB^2+BC^2
but by construction AD+DC=AC.
Substituting we have
AC⋅AC=AB^2+BC^2
This is equivalent to
AB^2+BC^2=AC^2
Hence the proof.
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