Need Help with a "Midpoint of a Line Segment" problem.

[KatieO]

This one has me stumped and I need someone's expertise.

"The base of an isosceles triangle lies on the x-axis. The coordinates of the midpoints of the equal sides of the triangle are (2,3) and (-2,3). What are the cooridnates of the vertices of the triangle?" Please help me out.

[Jane in NC]

This one has me stumped and I need someone's expertise.

"The base of an isosceles triangle lies on the x-axis. The coordinates of the midpoints of the equal sides of the triangle are (2,3) and (-2,3). What are the cooridnates of the vertices of the triangle?" Please help me out.

 

Two vertices lie on the x-axis. Let's call them (-x,0) and (x,0). The third vertex lies on the y axis, say (0,y). (I know by the symmetry of the midpoints that the vertices on the x axis are equidistant to the y axis.)

 

(If you drew a picture at this point, labeling those midpoints, I think you would "see" the solution.)

 

Now the midpoint of a line segment can be thought of as an average.

 

So we set up the equations:

 

(x + 0)/2 = 2 and (y + 0)/2 = 3

 

This gives x = 4; y = 6.

 

So you know that one vertex is (4,0); the others are (-4,0) and (0,6).

 

That's it.

 

Jane

[KatieO]

Thank you so much! I didn't consider the symmetry of the midpoints at all.