Given two line segments $\overline{AB} \text { and } \overline{CD}$ of lengths 7.5 cm and 4.6 cm respectively. Construct line segments.

(i) $\overline{PQ}$ of length equal to the sum of the lengths of $\overline{AB} \text { and } \overline{CD}$

(ii) $\overline{XY}$ of length equal to the difference of the lengths of $\overline{AB} \text { and } \overline{CD}$

Verify these lengths by measurements.

Steps:

1. Draw a line segment AB = 7.5 cm and a line segment CD = 4.6 cm using ruler and compass.

(i) Construction of PQ = AB + CD:

1. Draw any line l and mark a point P on it.

2. With the compass, take the length of AB (= 7.5 cm). Place the pointer at P and draw an arc cutting line l at point R. So, PR = AB = 7.5 cm.

3. Now, with the compass, take the length of CD (= 4.6 cm). Place the pointer at R and draw an arc cutting line l at point Q. So, RQ = CD = 4.6 cm.

Thus, PQ = PR + RQ = 7.5 cm + 4.6 cm = 12.1 cm.

On measuring PQ with a ruler, we find PQ = 12.1 cm.

Hence, the length of $\overline{PQ}$ = 12.1 cm.

(ii) Construction of XY = AB − CD:

1. Draw any line m and mark a point Z on it.

2. With the compass, take the length of AB (= 7.5 cm). Place the pointer at Z and draw an arc cutting line m at point Y. So, ZY = AB = 7.5 cm.

3. Now, with the compass, take the length of CD (= 4.6 cm). Place the pointer at Z and draw an arc cutting ZY at point X. So, ZX = CD = 4.6 cm.

Thus, XY = ZY − XZ = 7.5 cm − 4.6 cm = 2.9 cm.

On measuring XY with a ruler, we find XY = 2.9 cm.

Hence, the length of XY = 2.9 cm.