Mathematics

Problem

\triangle ABC

, two distinct lines each parallel to

BC

intersect

AB

and

AC

. The line nearer to

BC

meets

AB

D

and

AC

E

, while the line nearer to

A

meets

AB

F

and

AC

G

(with

F

between

A

and

D

). Given

AD = 6\,\text{cm}

DB = 12\,\text{cm}

AF = 4\,\text{cm}

and

AG = 5\,\text{cm}

, find the length of

GC

. a)

15.5\ \text{cm}

19.5\ \text{cm}

17.5\ \text{cm}

21.5\ \text{cm}

Intuitive explanation

The key is recognizing that parallel lines cutting the sides of a triangle create similar triangles, so corresponding ratios of segments are equal. Since $FG \parallel DE \parallel BC$ , triangle $AFG$ is similar to triangle $ADE$ , which is similar to triangle $ABC$ . By setting up the ratio $\frac{AF}{AD}$ and equating it to $\frac{AG}{AE}$ , you can find $AE$ , then use the same proportionality principle to find $GC$ .

Intuitive explanation

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