Question

assignment 5.1
triangle midsegments
find the value of x.
1
2
3
4
5
6

Explanation:

Step1: Apply the triangle mid - segment theorem

The triangle mid - segment theorem states that the length of a mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side.

Problem 1

• The third side is \(24\).
• By the mid - segment theorem, \(x=\frac{24}{2}\)
• \(x = 12\)

Problem 2

• The mid - segments of a triangle divide the sides proportionally. The sum of the lengths of the non - mid - segment parts of the sides related to \(x\):
• The mid - segments of a triangle, when considering the side related to \(x\), we know that \(x=5 + 7+8\)
• \(x = 20\)

Problem 3

• Let the side parallel to the mid - segment \(2x\). The length of the side parallel to the mid - segment:
• The length of the side parallel to \(2x\) is \(64\). By the mid - segment theorem, \(2x=\frac{64}{2}\)
• \(2x = 32\), then \(x = 16\)

Problem 4

• The side parallel to \(4x\) has length \(72\). By the mid - segment theorem, \(4x=\frac{72}{2}\)
• \(4x=36\), then \(x = 9\)

Problem 5

• The mid - segment is \(9\). By the mid - segment theorem, \(x - 7=2\times9\)
• \(x-7 = 18\), then \(x=18 + 7=25\)

Problem 6

• The mid - segment related to \(3x - 1\):
• The sum of the lengths of the two non - parallel sides of the trapezoid formed by the mid - segments: \(19+21\)
• By the mid - segment theorem for trapezoids (a special case of triangle mid - segments extended), \(3x-1=\frac{19 + 21}{1}\) (since the mid - segment of a trapezoid (formed by triangle mid - segments) is the average of the two non - parallel sides, but in this case of a triangle - related mid - segment structure, \(3x-1=19 + 21\))
• \(3x-1=40\), then \(3x=41\), \(x=\frac{41}{3}\approx13.67\)

Answer:

1. \(x = 12\)
2. \(x = 20\)
3. \(x = 16\)
4. \(x = 9\)
5. \(x = 25\)
6. \(x=\frac{41}{3}\)