Question

find the missing length indicated.

1. find ln
2. find rq
3. find sr
4. find vw

Explanation:

Step1: Use the mid - segment theorem for problem 13

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side.
In triangle \(LMN\), \(EF\) is the mid - segment. So \(EF=\frac{1}{2}LN\).
We have \(x + 2=\frac{1}{2}(x + 10)\)
Multiply both sides by 2: \(2(x + 2)=x + 10\)
Expand: \(2x+4=x + 10\)
Subtract \(x\) from both sides: \(2x - x+4=x - x+10\)
\(x+4 = 10\)
Subtract 4 from both sides: \(x=10 - 4=6\)
Then \(LN=x + 10\), substitute \(x = 6\) into it, \(LN=6 + 10=16\)

Step2: Use the mid - segment theorem for problem 14

In triangle \(SUT\), \(RQ\) is the mid - segment. So \(RQ=\frac{1}{2}SU\)
We know that \(x-2=\frac{1}{2}(x + 3)\)
Multiply both sides by 2: \(2(x - 2)=x + 3\)
Expand: \(2x-4=x + 3\)
Subtract \(x\) from both sides: \(2x - x-4=x - x+3\)
\(x-4 = 3\)
Add 4 to both sides: \(x=3 + 4=7\)
Then \(RQ=x - 2\), substitute \(x = 7\) into it, \(RQ=7 - 2=5\)

Step3: Use the mid - segment theorem for problem 15

In triangle \(EFD\), \(SR\) is the mid - segment. So \(SR=\frac{1}{2}FD\)
We have \(2x-14=\frac{1}{2}(x + 2)\)
Multiply both sides by 2: \(2(2x-14)=x + 2\)
Expand: \(4x-28=x + 2\)
Subtract \(x\) from both sides: \(4x - x-28=x - x+2\)
\(3x-28 = 2\)
Add 28 to both sides: \(3x=2 + 28=30\)
Divide both sides by 3: \(x = 10\)
Then \(SR=2x-14\), substitute \(x = 10\) into it, \(SR=2\times10-14=6\)

Step4: Use the mid - segment theorem for problem 16

In triangle \(MLN\), \(VW\) is the mid - segment. So \(VW=\frac{1}{2}MN\)
We know that \(x + 15=\frac{1}{2}(-2x-6)\)
Multiply both sides by 2: \(2(x + 15)=-2x-6\)
Expand: \(2x+30=-2x-6\)
Add \(2x\) to both sides: \(2x+2x+30=-2x+2x-6\)
\(4x+30=-6\)
Subtract 30 from both sides: \(4x=-6 - 30=-36\)
Divide both sides by 4: \(x=-9\)
Then \(VW=x + 15\), substitute \(x=-9\) into it, \(VW=-9 + 15=6\)

Answer:

1. \(LN = 16\)
2. \(RQ = 5\)
3. \(SR = 6\)
4. \(VW = 6\)