Question

What is the perimeter of the isosceles trapezoid that has vertices of `A(-3, 5), B(3, 5), C(5, -3),` and `D(-5, -3)`?

Answer 1

`16+2sqrt73`, or `33.088007`...

Explanation:

I would approach this problem in 3 steps:

1) Determine the length of the flat lines (the ones parallel to the `x`-axis),
2) Determine the length of the angled lines through the use of the Pythagorean Theorem, and
3) Find the sum of these values.

Let's start with the basic part: Determining the length of the flat lines.

You know that this trapezoid has 4 sides, and based of the coordinates, you know 2 of the sides are flat, and therefore easy to measure the length of.

In general, flat lines, or lines parallel with the `x`- or `y`-axes, have endpoints with either no change in `x` or no change in `y`.

In your case, there is no change in `y` for two lines.

These two lines are between points `A` and `B` (`(-3,5)` and `(3,5)`), and between points `C` and `D` (`(5,-3)` and `(-5,-3)`).

Both line `bar (AB)`'s length and line `bar (CD)`'s length can be found through their respective `Delta x` values.

For `bar (AB)`, `Delta x` would be `(3- -3)`, or `6`.
For `bar (CD)`, `Delta x` would be `(-5-5)`, or `-10`, but because distance is absolute you can simplify it to just `10`.

Next, we'll get the length of each of the slanted lines, which should conveniently be the same because this is an isosceles trapezoid.

We can achieve this through the use of the Pythagorean Theorem:

`a^2+b^2=c^2`,
Where:
`a` is the change in `x`,
`b` is the change in `y`, and
`c` is the length of the segment.

For sake of ease, we'll use line `bar (AD)`:

To get change in `x`, we'll use the equation `x_2-x_1=Deltax`.

Plug them in and you get:

`-5--3=-2`

We'll use a similar equation for change in `y`: `y_2-y_1=Deltay`

Again, plug and chug to get:

`-3-5=-8`

You now have your `a` and `b` values, so let's plug them into the Pythagorean Theorem:

`(-3)^2+(-8)^2=c^2`
`9+64=c^2`
`73=c^2`
`sqrt73=c`

Since we have the same line twice, but just reflected, we can use the same length twice.

For our final perimeter, we'll get:

`6 (bar (AB)) + 10 (bar (CD)) + 2*sqrt73 (bar (BC)+bar(DA))= 16+2sqrt73`

Which simplifies to:

`33.088007`...