Question 5

We will use properties of triangles and congruent shapes to show that the line segment AB cuts the line segment XY exactly in half and at a right angle.

Step 1 — Understanding the construction

Imagine we have a rope. Its ends are fixed at points X and Y. We use the rope to mark a point A. We stretch the rope so that the distance from X to A is the same as from Y to A. So, AX = AY. We then use the same rope length to mark another point B. We stretch the rope so that the distance from X to B is the same as from Y to B. And we use the same length of rope as before. So, BX = BY. This means all four lengths are equal.

$\boxed{\text{AX = AY = BX = BY}}$

Step 2 — Comparing two big triangles

Look at the two big triangles formed. These are triangle AXB and triangle AYB. We know AX is equal to AY. We also know BX is equal to BY. The line segment AB is a common side for both triangles. So, triangle AXB and triangle AYB have all three sides equal. This means they are congruent triangles. Congruent means they are identical in shape and size. Because they are congruent, their corresponding angles are also equal. So, the angle at A, angle XAB, is equal to angle YAB.

$\boxed{\text{Angle XAB = Angle YAB}}$

Step 3 — Comparing two smaller triangles

Let us call the point where line AB crosses line XY as point O. Now consider the two smaller triangles. These are triangle AXO and triangle AYO. We know AX is equal to AY from Step 1. We just found that angle XAO is equal to angle YAO. Angle XAO is the same as angle XAB. Angle YAO is the same as angle YAB. The line segment AO is a common side for both these triangles. So, triangle AXO and triangle AYO have two sides and the angle between them equal. This means they are also congruent triangles.

$\boxed{\text{Triangle AXO is congruent to Triangle AYO}}$

Step 4 — AB cuts XY in half

Since triangle AXO and triangle AYO are congruent, their corresponding sides are equal. This means the side OX is equal to the side OY. So, point O is exactly in the middle of the line segment XY. This proves that AB bisects XY.

$\boxed{\text{OX = OY}}$

Step 5 — AB meets XY at a right angle

Since triangle AXO and triangle AYO are congruent, their corresponding angles are equal. So, angle XOA is equal to angle YOA. These two angles, angle XOA and angle YOA, form a straight line on XY. Angles on a straight line add up to 180 degrees. Since they are equal and add up to 180 degrees, each angle must be half of 180 degrees. So, angle XOA = 90 degrees. Angle YOA also equals 90 degrees. This means AB is perpendicular to XY.

$\boxed{\text{Angle XOA = 90 degrees}}$

Step 6 — Conclusion

From Step 4, we know AB bisects XY. From Step 5, we know AB is perpendicular to XY. Therefore, AB is the perpendicular bisector of XY.

Answer

In the above figure, XAY and XBY are two positions of the rope. Points A and B are at the midpoint of the rope. Therefore, we have AX = AY = BX = BY. Triangle AXB and Triangle AYB are congruent, because AX = AY, BX = BY, and AB is common. Therefore, angle XAO = angle YAO. Triangle AXO and Triangle AYO are congruent, because AX = AY, angle XAO = angle YAO, and AO is common. Therefore, OX = OY and angle XOA = angle YOA. Also, angle XOA + angle YOA = 180 degrees. Therefore, 2 * angle XOA = 180 degrees or angle XOA = 90 degrees. Therefore, OX = OY and angle XOA = angle YOA = 90 degrees. Therefore, by definition, AB is the perpendicular bisector of the line XY.