In $\triangle XYZ$, XY = 6 cm. If M and N are two points on XY and XZ respectively such that MN $\|$ YZ and XN = $\frac{1}{4}$ XZ, then the length of XM is :

Step 1: Identify similar triangles

Given that MN is parallel to YZ, by the Basic Proportionality Theorem (also known as Thales's Theorem or the Intercept Theorem), the line segment MN divides the sides XY and XZ proportionally. This also implies that triangle XMN is similar to triangle XYZ.

Step 2: Use similarity ratio

Since $\triangle XMN$ is similar to $\triangle XYZ$, the ratio of their corresponding sides must be equal. Therefore, the ratio of XM to XY is equal to the ratio of XN to XZ.

Step 3: Substitute given values

We are given that XY = 6 cm and XN = $\frac{1}{4}$ XZ. Substituting these values into the similarity ratio equation, we get $\frac{XM}{6} = \frac{1}{4}$.

Step 4: Solve for XM

To find the length of XM, we multiply both sides of the equation by 6. This simplifies to XM equals 6 multiplied by one-fourth, which is 1.5 cm.

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