Question 37916: Hi, I am having trouble trying to figure out how to do this problem. I am in 9th grade geometry but we are do a review of Algebra/Coordinate Proof. The question is: Position and label a right isosceles triangle on the coordinate plane. Then prove that the segment joining the midpoint of the two legs of the right triangle is parallel to the hypotenuse. Thanks so much for your help- Will
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! Probably the easiest way to see this is to draw the right isosceles triangle with the right angle at the origin of your coordinate axes. Then draw the horizontal leg a distance of a-units along the positive x-axis and the vertical leg a distance of a-units along the positive y-axis. Connect the ends of the x-axis leg and the y-axis leg to form the hypotenuse.
You can label the triangle with A at the right-angle vertex, B at the end of the x-axis leg, and C at the end of the y-axis leg.
With this labeling, the hypotenuse is segment CB, the x-axis leg is AB, and the y-axis leg is AC.
Now, place a point (E) on the x-axis leg at a distance of a/2 from the origin (this point bisects the segment AB) and another point (F) on the y-axis leg at a distance of a/2 from the origin (this point bisects the segment AC), then connect these two points.
The proof:
Using the fact that parallel lines have identical slopes, find the slope of the hypotenuse. Slope is rise over run and for the hypotenuse, the rise is distance a and the run is distance a so the slope is a/a = 1 (it's really a negative slope).
The slope of the line connecting the midpoints of the two legs is (a/2)/(a/2) = 1 (again, this is really a negative slope)
Since the slopes of the two lines are equal, the lines are parallel. QED
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