Given:The figures of triangles and their mid segments.To find:The values of n.Solution:Mid-segment theorem: According to this theorem, mid segment of the triangle is a line segment that bisect the two sides of the triangle and parallel to third side, The measure of mid-segment is half of the parallel side.9. It is given that:Length of mid-segment = 54Length of parallel side = 3nBy using mid-segment theorem for the given triangle, we get[tex]54=\dfrac{1}{2}(3n)[/tex][tex]2\times 54=3n[/tex][tex]108=3n[/tex]Divide both side by 3.[tex]\dfrac{108}{3}=n[/tex][tex]36=n[/tex]Hence, the value of n is equal to 36.10. It is given that:Length of mid-segment = 4n+5Length of parallel side = 74By using mid-segment theorem for the given triangle, we get[tex]4n+5=\dfrac{1}{2}(74)[/tex][tex]4n+5=37[/tex][tex]4n=37-5[/tex][tex]4n=32[/tex]Divide both side by 4.[tex]n=\dfrac{32}{4}[/tex][tex]n=8[/tex]Hence, the value of n is equal to 8.