You have isosceles triangle WXY and segment WZ is the perpendicular bisector of segment XY. You have to prove triangle WXY isosceles in only three steps using the theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment. Can someone please help me?
Sure! I can help you prove that triangle WXY is isosceles using the given theorem. Let's go step by step:
Step 1: Given that segment WZ is the perpendicular bisector of segment XY.
Step 2: According to the theorem "If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment," we can conclude that point Z, which lies on the perpendicular bisector WZ, is equidistant from the endpoints X and Y of segment XY.
Step 3: Since point Z is equidistant from X and Y, we can conclude that segments ZW and ZY are congruent (have equal lengths) because they are radii of the same circle with Z as its center.
Now, we can establish that triangle WZY is an isosceles triangle because it has two congruent sides, ZW and ZY.
Since triangle WZY is isosceles, we can use the Isosceles Triangle Theorem, which states that the angles opposite the congruent sides are also congruent. Therefore, angle WYZ is congruent to angle WZY.
Since angles WYZ and WZY are congruent, we conclude that base angle WXY is also congruent to angle WYZ (because the sum of angles in a triangle is 180 degrees). Therefore, triangle WXY is an isosceles triangle.
By following these three steps and applying the given theorem, we have successfully proved that triangle WXY is isosceles.