Question

find the missing length indicated.

1. find ( ik )

Explanation:

Step1: Use the mid - segment theorem

In a triangle, if a line segment connects the mid - points of two sides of a triangle, then it is parallel to the third side and its length is half of the length of the third side. Here, \(EF\) is the mid - segment of \(\triangle ILK\) (since \(E\) is the mid - point of \(IL\) as \(IE = EL\)). So, \(EF=\frac{1}{2}LK\).

Step2: Set up the equation

We know that \(EF = 2x + 22\) and \(LK=(x + 11)+(x + 11)=2(x + 11)\). According to the mid - segment theorem \(2x+22=\frac{1}{2}\times(2x + 26)\)

First, simplify the right - hand side of the equation: \(\frac{1}{2}\times(2x + 26)=x + 13\)

So the equation becomes \(2x+22=x + 13\)

Subtract \(x\) from both sides: \(2x - x+22=x - x + 13\), which gives \(x+22 = 13\)

Subtract 22 from both sides: \(x=13 - 22=-9\)

Step3: Calculate \(LK\)

Since \(LK = 2(x + 11)\), substitute \(x=-9\) into the formula.

\(LK=2(-9 + 11)=2\times2 = 4\)

Answer:

\(LK = 4\)