mind your decisions I'm presto Walker
inside a semicircle two side-by-side
squares are inscribed if the semi
circles radius is equal to 10 can you
solve for the total area of the two
squares if no demonstrate why not
if yes calculate the answer this problem
is adapted from math stack exchange be
sure to consider all possible ways that
two side by side squares could be
inscribed in the semicircle pause if
you'd like to give this problem a try
and when you're ready keep watching to
learn how to solve this problem
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in a right triangle with legs of a and B
and a hypotenuse C we all know that a
squared plus B squared is equal to C
squared but what should this rule be
called according to mathematician Munjal
Bhargav who's a Fields Medal winner the
Pythagorean theorem is clearly the wrong
name it should either be an Egyptian
theorem if you look at the standard of
just having an idea about it an Indian
theorem if you're looking for a complete
statement of it or a Chinese theorem if
you're looking for the proof of it
I'm preferential towards the proof of it
and I suggest we call it the Gogu
theorem which is what the theorem is
known as in China getting the history
correct is very important as manju
bhargavi explains knowing the correct
history of mathematics was useful in my
own research because if you learned from
the original source how an idea came
about that can give you great insight
now let's solve the problem suppose one
square has a side length of X and the
other square has a side length of Y the
total area of the two squares will be
equaled x squared plus y squared draw
the diagonal of the X square its
diagonal length will be equal to x times
the square root of 2 and the diagonal
will bisect the right angle into two 45
degree angles now draw the other
diagonal its length will be equal to Y
times the square root of 2 and it will
form a 45-degree angle right here thus
the two diagonals form a 90 degree right
angle draw a cord connecting these
diagonals and call it Z this is a right
triangle so we can use the Gogu theorem
to deduce the length of z squared this
simplifies to be 2x squared plus 2y
squared we'll now figure out the measure
of this circular arc we'll focus on this
diagram and reflect the entire diagram
across the diameter what is the measure
of this inscribed angle it will bisect
the right angle of the square therefore
it'll be equal to 45 degrees we can thus
calculate
the measure of this circular arc by the
inscribed angle theorem
it'll be equal to two times 45 degrees
or 90 degrees how does this help us
we'll go back to our diagram and
consider another way to calculate this
chord Z from the center of the circle
draw radii to the endpoints of the chord
since this circular arc is 90 degrees
this central angle will be equal to 90
degrees as well
it'll be a right angle we can thus apply
the Gogu theorem again to get Z squared
is equal to R squared plus R squared we
now have two equations which will
simplify the second equation simplifies
to be two R squared and since both
equations are equal to C squared we set
them equal to each other we then divide
by 2 to get x squared plus y squared is
equal to R squared thus the total area
will be equalled R squared if R is equal
to 10 it's equal to 10 squared which is
equal to 100 and that's our answer
regardless of how the two side by side
squares are inscribed incredible thanks
for making mind your decisions one of
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support
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