Question

Hiep is writing a coordinate proof to show that the midsegment of a trapezoid is parallel to its bases. He starts by assigning coordinates as given, where RS¯¯¯¯¯ is the midsegment of trapezoid KLMN. am I correct?

Answer 1

Please see below.

Explanation:

.

If `(x_1,y_1)` and `(x_2,y_2)` are coordinates of the end points of a line segment the coordinates of the midpoint can be found using the following formulas:

`((x_1+x_2)/2,(y_1+y_2)/2)`

In order to prove that `RS` is the midsegment of the trapezoid, we need to prove that `R` is the midpoint of `KN`, and `S` is the midpoint of `LM` and `RS` is parallel to the bases.

The coordinates of `R` and `S` are:

`R((2b+0)/2,(2c+0)/2)=R(b,c)`

`S((2a+2d)/2,(2c+0)/2)=S(a+d,c)`

`Slope_(RS)=m=(y_2-y_1)/(x_2-x_1)=(c-c)/(a+d-b)=0/(a+d-b)=0`

`Slope_(NM)` and `Slope_(KL)` are also `0`

Since the all three slopes are equal, all three lines are parallel to each other.

Therefore, `RS` is midsegment of the trapezoid.