In △XYZ, XY = 6 cm. If M and N are two points on XY and XZ respectively such that MN ∥ YZ and XN = 41 XZ, then the length of XM is :
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Step-by-Step Solution
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 △XMN is similar to △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 = 41 XZ. Substituting these values into the similarity ratio equation, we get 6XM=41.
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.