Question

1. find the value of x

19x - 36
6x + 31

Explanation:

Step1: Identify Midline Theorem

The segment \(6x + 31\) is a midline of the triangle (since it connects midpoints of two sides), so it's half the length of the base \(19x - 36\). Thus, \(6x + 31=\frac{1}{2}(19x - 36)\) or (alternatively, if it's a trapezoid with midline, but here triangle midline: midline = \(\frac{1}{2}\) base). Wait, actually, looking at the markings, the sides are bisected, so the segment is a midline, so \(6x + 31=\frac{19x - 36}{2}\)? Wait, no, maybe it's a trapezoid? Wait, no, the figure: the two sides with single marks are bisected, and the diagonal with double marks are bisected. Wait, actually, the segment \(6x + 31\) is parallel to \(19x - 36\) and connects midpoints, so by Midline Theorem (Triangle Midline Theorem), the midline is half the length of the third side. Wait, no, maybe it's a trapezoid with midline? Wait, no, the figure: let's re-express. Wait, the correct approach: since the segment connects midpoints (as per the markings: single marks on one side, double on the diagonal, so midpoints), so the length of the midline is half the length of the base. Wait, no, actually, in a triangle, midline is parallel to the base and half its length. So \(6x + 31=\frac{1}{2}(19x - 36)\)? Wait, no, maybe I got it reversed. Wait, no, maybe the segment \(6x + 31\) is the midline, so \(2(6x + 31)=19x - 36\)? Wait, no, that would be if the midline is half, so base is twice the midline. Wait, let's check: Midline Theorem states that the midline is parallel to the third side and half as long. So if the midline is \(6x + 31\), then the third side (base) is \(2(6x + 31)\). But in the figure, the base is \(19x - 36\), so:

\(19x - 36 = 2(6x + 31)\)

Step2: Solve the Equation

Expand the right side: \(19x - 36 = 12x + 62\)

Subtract \(12x\) from both sides: \(19x - 12x - 36 = 62\) → \(7x - 36 = 62\)

Add 36 to both sides: \(7x = 62 + 36\) → \(7x = 98\)

Divide by 7: \(x = \frac{98}{7}=14\)

Answer:

\(x = 14\)