Question

part v. fill in the blanks and solve the following problems. show your solutions. complete solution will get 5 points. direct answer gets only 2.

1. mid - segment theorem - the mid - segment of a triangle is ______ the side parallel to it.

triangle sit is shown, where ( overline{mg} ) is the midsegment. ( sl = 2x + 6 ), ( st = 3x + 16 ), ( lt = 8x - 14 ), and ( mg = 3x - 5 ).

Explanation:

Step1: Use the property of mid - segment in a triangle

In a triangle, the mid - segment (a segment connecting the mid - points of two sides of a triangle) is parallel to the third side and half its length. So, the mid - segment of a triangle is half the side parallel to it.

Step2: Use the mid - segment formula \(MG=\frac{1}{2}LT\)

Given \(SL = 2x+6\), \(ST=2x + 16\), \(LT=8x-14\), and \(MG = 3x-5\)
Since \(MG=\frac{1}{2}LT\) (mid - segment theorem), we substitute the expressions:
\(3x - 5=\frac{1}{2}(8x-14)\)
Multiply both sides by 2 to get rid of the fraction:
\(2(3x - 5)=8x-14\)
Expand the left - hand side: \(6x-10 = 8x-14\)
Subtract \(6x\) from both sides: \(-10=2x-14\)
Add 14 to both sides: \(4 = 2x\)
Divide both sides by 2: \(x = 2\)

Step3: Find the length of \(MG\)

Substitute \(x = 2\) into the expression for \(MG\):
\(MG=3x-5\)
\(MG=3\times2-5\)
\(MG=6 - 5\)
\(MG = 1\)

Answer:

The mid - segment of a triangle is half the side parallel to it. The value of \(MG\) is \(1\).