Question

i \can\ similarity solve for x. triangle with labels: l, m, n; segments: lm = x + 3, mn = 10, lp = x, pn = 6 a 8.25 b 5 c 4.5 d 7.5

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. So, \(\frac{LM}{MN}=\frac{LP}{PN}\).

Step2: Substitute the given values

Given \(LM = x + 3\), \(MN=10\), \(LP = x\), and \(PN = 6\). Substituting into the formula \(\frac{x + 3}{10}=\frac{x}{6}\).

Step3: Cross - multiply

Cross - multiplying gives \(6(x + 3)=10x\).

Step4: Expand and solve for \(x\)

Expand \(6(x + 3)\) to get \(6x+18 = 10x\). Then, \(18=10x - 6x\), \(18 = 4x\), \(x=\frac{18}{4}=4.5\).

Answer:

C. 4.5