We are given the line segment GH, and we are asked to construct a perpendicular line that passes through the midpoint of GH, M. To do that, we need to place a compass in point G and draw a circle of radius GH, that is we place the end of the compass at point H. Then we need to place the compass at point H and draw a circle of radius GH, that is the other end of the compass must be at point G. It looks like this:
Now we draw a line joining the points where the two circles intercept, like this:
The resulting line is perpendicular to GH and passes through the midpoint M. We mark one of the endpoints of the new line as K. When joining HK and GK we get the following:
To map triangle GMK into triangle HMK we would have to do a reflection (which is a rigid motion) across the line KM.
GK and HK are congruent or they have the same measure because triangles GMK and HMK are right tringles since lines KM and GH are perpendicular, moreover, since M is a midpoint that means that segments GM and MH have the same length and since both triangles share the side KM, that means that sides GK and KH must also be the same.
