Welcome to two examples,
on how to use the Pythagorean Theorem
to determine the distance between two points
on the coordinate plane.
We're asked to find the distance between
point A and point D which would be this distance,
as well as the distance between point B and point E,
which'd be this distance.
In order to use the Pythagorean Theorem
to determine the distance between two points
on the coordinate plane,
we'll form right triangles on the coordinate plane,
where the distance between the two points,
would by the hypotenuse of the right triangle.
So looking at the distance between point A and point D,
we'll form a right triangle where one leg would
represent the vertical distance between point A and point D.
This distance here.
The other leg would represent the horizontal distance
between point A and point D,
which would be this distance here.
Notice now we have a right triangle
on the coordinate plane,
and we can also determine the length
of these two sides that form the right angle,
which are the two legs of the right triangle.
This side is the length of one, two, three, four units.
This side is the length
of one, two, three, four, five units.
The unknown length is a hypotenuse,
and again the unknown length is the hypotenuse.
The Pythagorean Theorem is
A squared plus B squared equals C squared,
or C squared equals A squared plus B squared,
where C is the length of the hypotenuse,
the longest side of the right triangle.
It's also the side opposite the right angle.
And A and B, the length of the two legs,
where the legs form the right angle.
Looking at this right triangle,
C, the hypotenuse, is the unknown length.
Let's go ahead and let A equal four,
and B equal five.
So using the Pythagorean Theorem,
C squared equals A squared plus B squared.
We would have C squared equals,
A squared would be four squared,
plus B squared would be five squared.
So we have C squared equals 16 plus 25.
So C squared equals 41.
To solve for C, we will now take the square root
of both sides of the equation.
But here, because we know C is a length,
we know C must be positive,
and therefore when we square root
both sides of the equation,
the square root of of C squared is equal to C,
again because you know C is positive.
So C equals the square root of 41.
And now we'll use the calculator
to round this value to the nearest 10th.
So if we press second X squared,
it brings up the square root,
and we enter 41, and then enter.
So one decimal place or the 10ths,
C is approximately 6.4 units.
So now we know this length is approximately 6.4 units.
So we'll say the distance between
point A and point D, is approximately
6.4 units.
And now we'll find the distance between point B and point E.
So we'll form another right triangle,
where this side would be the hypotenuse.
So one leg would give us the vertical distance
between point B and point E, this distance here.
The other leg would represent the horizontal distance
between point B and point E,
which would be this distance here.
Again we have a right triangle on the coordinate plane.
This leg would have a length of
one, two, three, four, five, six units.
This leg has a length of one, two, three, four,
five, six, seven, eight units.
And this distance here, the hypotenuse,
has an unknown length.
Let's label this leg A, this leg B,
We know the hypotenuse must be C.
So using the Pythagorean Theorem,
C squared equals A squared plus B squared,
we would have C squared equals
six squared plus eight squared.
So C squared equals 36 plus 64.
So, C squared is equal to, 36 plus 64 is 100.
Again, because C is a length, C must be positive.
So when we take the square root
of both sides of the equation,
we get C equals exactly 10 units.
So now we know the distance between B and E,
is 10 units.
I hope you found this helpful.