Question

what is the distance between points m and n?
\boxed{} meters
(diagram: triangle/figure with points j, k, l, m, n; jk = 19.6 m, km = 10.7 m, ml = 10.7 m, jn = 7.4 m, nl = 7.4 m)

Explanation:

Step1: Analyze the segment LN and JN

From the diagram, we can see that \( JN = 7.4 \) m and \( NL = 7.4 \) m, and also \( KM = 10.7 \) m and \( ML = 10.7 \) m. This suggests that triangle \( JKL \) is isoceles and \( N \) and \( M \) are mid - points of \( JL \) and \( KL \) respectively. By the mid - segment theorem (or the property of similar triangles), the segment \( MN \) should be parallel to \( JK \) and \( MN=\frac{1}{2}JK \). But we can also directly observe the length of the segments. Wait, actually, looking at the segments \( JL \): \( JN = 7.4\) m and \( NL = 7.4\) m, so \( JL=JN + NL=7.4 + 7.4 = 14.8\) m? No, wait, no. Wait, the key here is that the length of \( MN \) can be found by noticing that the triangle is symmetric and the length of \( MN \) is equal to the length of \( JN \)? No, wait, no. Wait, let's look at the other side. The length from \( K \) to \( M \) is \( 10.7\) m and from \( M \) to \( L \) is \( 10.7\) m, so \( KL=10.7 + 10.7=21.4\) m. But for \( JL \), \( JN = 7.4\) m and \( NL = 7.4\) m, so \( JL = 7.4+7.4 = 14.8\) m. But the length of \( JK \) is \( 19.6\) m. Wait, maybe a better way: since \( N \) is the mid - point of \( JL \) (because \( JN=NL = 7.4\) m) and \( M \) is the mid - point of \( KL \) (because \( KM = ML=10.7\) m), then by the mid - segment theorem in a triangle, the segment connecting the mid - points of two sides of a triangle is parallel to the third side and half its length. So in triangle \( JKL \), \( MN \) connects the mid - points of \( JL \) and \( KL \), so \( MN=\frac{1}{2}JK \). Wait, but \( JK = 19.6\) m, so \( MN=\frac{19.6}{2}=9.8\) m? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, looking at the diagram again, maybe the triangle is such that \( JL \) and \( KL \) have segments of \( 7.4\) and \( 10.7\) respectively, and \( MN \) is equal to \( 7.4\) m? No, wait, no. Wait, the problem is to find the distance between \( M \) and \( N \). Let's look at the lengths: \( JN = 7.4\) m, \( KM = 10.7\) m, but maybe the figure is a parallelogram? No, it's a triangle. Wait, another approach: the length of \( MN \) is equal to \( 7.4\) m? No, wait, no. Wait, let's check the given lengths. The segment \( JL \) is divided into two equal parts: \( JN = 7.4\) and \( NL = 7.4\). The segment \( KL \) is divided into two equal parts: \( KM = 10.7\) and \( ML = 10.7\). So triangle \( JKL \) has \( JK = 19.6\), \( JL=14.8\), \( KL = 21.4\). Then by the mid - segment theorem, the mid - segment \( MN \) (connecting mid - points of \( JL \) and \( KL \)) should be parallel to \( JK \) and \( MN=\frac{1}{2}JK\). So \( JK = 19.6\), so \( MN=\frac{19.6}{2}=9.8\)? Wait, but that doesn't seem right. Wait, maybe the figure is a trapezoid? No, it's a triangle. Wait, maybe I misread the diagram. Wait, the points are \( J\), \( K\), \( L\) forming a triangle, with \( N\) on \( JL\) and \( M\) on \( KL\). \( JN = 7.4\), \( NL = 7.4\), \( KM = 10.7\), \( ML = 10.7\), \( JK = 19.6\). So using the mid - segment theorem: in triangle \( JKL\), if \( N\) is the mid - point of \( JL\) (\(JN = NL\)) and \( M\) is the mid - point of \( KL\) (\(KM = ML\)), then \( MN\parallel JK\) and \( MN=\frac{1}{2}JK\). So \( MN=\frac{19.6}{2}=9.8\) meters. Wait, but let's check with another method. The length of \( JL\) is \( 7.4 + 7.4=14.8\) m, length of \( KL\) is \( 10.7+10.7 = 21.4\) m, length of \( JK = 19.6\) m. Using the formula for the mid - segment: yes, mid - segment length is half the length of the third side. So \( MN=\frac{1}{2}\times19.6 = 9.8\) meters.

Step2: Calculate the length of MN

Given that \( N\) is the mid - point of \( JL\) and \( M\) is the mid - point of \( KL\) in \(\triangle JKL\). By the mid - segment theorem, the length of the segment connecting the mid - points of two sides of a triangle is half the length of the third side. Here, the third side is \( JK = 19.6\) m. So \( MN=\frac{1}{2}\times JK\)
\[MN=\frac{19.6}{2}=9.8\]

Answer:

\(9.8\)