Question

Given the three topics listed below, discuss a visual, verbal, and algebraic way of connecting the concepts:

1. The distance formula

2. The standard equation of a circle (not centered at the origin)

3.The Pythagorean Theorem

Can you think of other related mathematics topics that can be extended from these three? Write a 1/2 page not including diagrams.

Answer 1

Lots of connections!

Explanation:

I attached some images to make it more clear.

Visual and verbal discussion:

A good starting point is the circle. One can think of a circle as the set of points that are equidistant to a certain point. In that sense, one can define a circle using the distance. At the same time, given a point
`(x_(0),y_(0))` in the plane, we can connect the point with the origin of the coordinates system forming a rectangle triangle! (See 2nd image)

Algebraic discussion:

1. The distance Formula:

Given two points in the plane
`(x_(1),y_(1))` and
`(x_(2),y_(2))` we can find the distance between both points with the distance formula:

`d = \sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2}`

2. Equation of a circle centered at a point
`(x_(0),y_(0))`

`r = \sqrt{(x-x{_0}^2) + (y-y_(0))^2}`

3. The Pythagorean Theorem:

Given a triangle of sides
`a;b` and hypotenuse
`h` we have that

`h^2 = a^2 + b^2`

Only by writing this equations we can already see the similarities between all three.

In fact, the most amazing thing about all three is that they are equivalents. That is, we can obtain every single one of them as an immediate result of another.

For instance, as I said before, one can think of a circle as the set of points equidistant to a certain origin or center point. So we could use the distance formula for each point on the circumference and we would obtain always the same value
`r` hence obtaining the equation of a circle.

Also as we discussed, any point on the plane form a rectangle triangle with its coordinates and calculating the distance of said point to the origin of the coordinate system would give us no other thing than the hypotenuse of said triangle!