Use the following figure to find :

(i) BC, if AB = 7.2 cm.

(ii) GE, if FE = 4 cm.

(iii) AE, if BD = 4.1 cm.

(iv) DF, if CG = 11 cm.

By equal intercept theorem,

If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.

(i) From figure,

CG || BF || AE

Since, CD = DE

∴ BC = AB = 7.2 cm (By equal intercept theorem)

Hence, BC = 7.2 cm.

(ii) From figure,

CG || BF || AE

Since, CD = DE

∴ FG = FE = 4 cm (By equal intercept theorem)

From figure,

⇒ GE = FG + FE = 4 + 4 = 8 cm.

Hence, GE = 8 cm.

(iii) By mid-point theorem,

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.

By converse of mid-point theorem,

The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.

In △ AEC,

D is mid-point of CE

⇒ BF || AE

∴ BD || AE.

∴ B is mid-point of AC. (By converse of mid-point theorem)

∴ BD = $\dfrac{1}{2}AE$ (By mid-point theorem)

⇒ AE = 2BD = 2 × 4.1 = 8.2 cm

Hence, AE = 8.2 cm.

(iv) In △ EGC,

D is mid-point of CE.

⇒ BF || CG

∴ DF || CG.

∴ F is mid-point of GE. (By converse of mid-point theorem)

∴ DF = $\dfrac{1}{2}CG$ (By mid-point theorem)

⇒ DF = $\dfrac{1}{2} \times 11$ = 5.5 cm

Hence, DF = 5.5 cm.