Question

In triangle ABC shown below, point A is at (0,0)

point B is at (x subscript2, 0)
point C is ([x sub1]/ 2, [y sub1]/2
point D is [x sub1 + xsub2]/2, ysub1/2)

Prove that segment DE is parallel to segment AB.

Answer 1

Proof below

Explanation:

To prove DE and AB are parallel, we compare the slopes of both segments

Suppose we know a line passes through points A(x1,y1) and B(x2,y2). The slope can be calculated with the equation:

`\displaystyle m=(y_2-y_1)/(x_2-x_1)`

Segment AB goes through the points (0,0) and (x2,0), thus:

`\displaystyle m_(AB)=(0-0)/(x_2-0)=0`

The slope is 0 because the line is horizontal.

Now for the segment DE, the endpoints are

`\displaystyle ((x_1)/(2),(y_1)/(2)),\ ((x_1+x_2)/(2),(y_1)/(2))`

The slope is:

`\displaystyle m_(DE)=((y_1)/(2)-(y_1)/(2))/((x_1+x_2)/(2)-(x_1)/(2))`

`\displaystyle m_(DE)=(0)/((x_2)/(2))=0`

Segment DE is also horizontal, thus is parallel to segment AB.