Come up with a method to construct the angle bisector using a rope.

Question

Homi · Accepted Answer

We will find a point inside the angle that is equally far from both arms.

Step 1 — Mark points on the arms

Let us start with an angle. We call it angle XOY. Point O is the corner of the angle. Let us place a small pole at point O. Take a rope. Make a loop at one end of the rope. Put this loop around the pole at O. Hold the rope tight along the arm OX. Mark a point on the rope. This is a fixed length. Let us call this length $L_1$ . Mark a point A on arm OX at this length $L_1$ from O. Now, swing the rope to arm OY. Keep the rope length $L_1$ the same. Mark a point B on arm OY at this length $L_1$ from O. So, the distance from O to A is the same as O to B.

$OA = L_1$

$OB = L_1$

$\boxed{OA = OB}$

Diagram 1

Step 2 — Find a midpoint

Now, place small poles at point A and point B. Take another rope. Make loops at both ends. Fix one loop to the pole at A. Fix the other loop to the pole at B. Hold this rope exactly in its middle. Let us call this middle point M. This means the distance from A to M is the same as B to M. Let us call this length $L_2$ .

$AM = L_2$

$BM = L_2$

$\boxed{AM = BM}$

Step 3 — Draw the bisector

Now, connect point O to point M with a straight line. This line is OM. Consider the triangle OAM. Consider the triangle OBM. We know that OA equals OB from Step 1. We know that AM equals BM from Step 2. The side OM is common to both triangles. So, triangle OAM and triangle OBM are identical. This is because of the SSS (Side-Side-Side) rule. If the triangles are identical, their angles are also identical. So, angle AOM is equal to angle BOM. This means the line OM divides angle XOY into two equal parts. So, OM is the angle bisector of angle XOY.

$\boxed{\text{OM is the angle bisector}}$

Answer

(i) Fix a rope at the angle's corner O. Mark points A and B on the arms at equal distances from O. (ii) Fix another rope between points A and B. Find its exact middle point, M. (iii) The line connecting O to M is the angle bisector.

Question 11

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