Question The points \( \mathrm{K}, \mathrm{L}, \mathrm{M} \) and N all lie on the same line segment, in that order, such that the ratio of \( K L: L M: M N \) is equal to \( 2: 4: 3 \). If \( K N=36 \), find \( L N \). Answer Attempt 1 out of 3 \( L N=\square \) Submit Answer

To solve the problem, we first need to assign variables to the segments based on the given ratio of \( KL:LM:MN = 2:4:3 \). Let's denote the lengths of these segments as follows: let \( KL = 2x \), \( LM = 4x \), and \( MN = 3x \). The total length \( KN \) can be expressed as the sum of these segments: \[ KN = KL + LM + MN = 2x + 4x + 3x = 9x \] Given that \( KN = 36 \), we can set up the equation: \[ 9x = 36 \] Solving for \( x \) gives us: \[ x = 4 \] Now we can find the lengths of \( KL \), \( LM \), and \( MN \): \[ KL = 2x = 2(4) = 8, \quad LM = 4x = 4(4) = 16, \quad MN = 3x = 3(4) = 12 \] Now, we need to calculate \( LN \): \[ LN = LM + MN = 16 + 12 = 28 \] Thus, the final answer is: \[ L N = 28 \]