Question

find the length of \\(\overline{lk}\\). \\(\triangle hjk\\) with a segment from \\(j\\) to \\(l\\) on \\(hk\\). \\(jh = 21\\), \\(jk = 2y - 4\\), \\(hl = 14\\), \\(lk = y + 1\\). options: a 9, b 7, c 15, d 12

Explanation:

Step1: Use the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. So, \(\frac{HL}{LK}=\frac{JH}{JK}\). Substitute \(HL = 14\), \(JH=21\), \(JK = 2y - 4\), and \(LK=y + 1\) into the proportion: \(\frac{14}{y + 1}=\frac{21}{2y-4}\).

Step2: Cross - multiply

Cross - multiplying gives \(14\times(2y - 4)=21\times(y + 1)\). Expand both sides: \(28y-56 = 21y+21\).

Step3: Solve for \(y\)

Subtract \(21y\) from both sides: \(28y-21y-56=21y - 21y+21\), which simplifies to \(7y-56 = 21\). Add \(56\) to both sides: \(7y=21 + 56\), so \(7y=77\). Divide both sides by \(7\): \(y = 11\).

Step4: Find the length of \(LK\)

Since \(LK=y + 1\), substitute \(y = 11\) into the expression. Then \(LK=11 + 1=12\).

Answer:

D. 12