In triangle KLM, points D and E lie on sides KL and KM respectively, such that DE ∥ LM. If KD = 4 cm, DL = 12 cm, and KE = 6 cm, find the length of EM.

The correct option is 18 cm.

Step-by-step explanation:

According to the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

In triangle KLM, line segment DE is parallel to side LM (DE ∥ LM), with points D and E lying on sides KL and KM respectively. Therefore, by the Basic Proportionality Theorem:

$\frac{\mathrm{KD}}{\mathrm{DL}}=\frac{\mathrm{KE}}{\mathrm{EM}}$

We are given the following values:

KD = 4 cm
DL = 12 cm
KE = 6 cm

Substitute the given values into the proportion equation:

$\frac{4}{12}=\frac{6}{\mathrm{EM}}$

Simplify the fraction on the left side:

$\frac{1}{3}=\frac{6}{\mathrm{EM}}$

Cross-multiply to solve for EM:

$\mathrm{EM}=6\times 3$

$\mathrm{EM}=18\text{cm}$

Thus, the length of EM is 18 cm.