A city park designer is designing a new park. The park will be shaped like aright triangle and there will be two pathways for pedestrians, shown by VTand VW in the diagram. The park planner only wrote two lengths on hissketch as shown. Based on the diagram, what will be the lengths of the twopathways?

Answer:Part a) [tex]VW=12\ yd[/tex]Part b) [tex]VT=9\ yd[/tex]Step-by-step explanation:The complete question in the attached figurewe know thatThe Triangle Mid-segment Theorem  states that:The segment joining the midpoints of two sides of a triangle is parallel to the third  side, and its length is half the length of that sidestep 1Find out the length of the pathway VW[tex]VW=\frac{XW}{2}[/tex] -----> by Triangle Mid-segment TheoremBecause V is the midpoint of XY and W is the midpoint of YZsoVW is parallel to XWwe have[tex]XW=24\ yd[/tex]substitute [tex]VW=\frac{24}{2}=12\ yd[/tex]step 2Find the length YZApplying the Pythagoras Theorem[tex]XY^2=XZ^2+YZ^2[/tex]we have[tex]XY=30\ yd\\XZ=24\ yd[/tex]substitute[tex]30^2=24^2+YZ^2[/tex][tex]YZ^2=30^2-24^2[/tex][tex]YZ^2=324[/tex][tex]YZ=18\ yd[/tex]step 3Find out the length of the pathway VT[tex]VT=\frac{YZ}{2}[/tex] -----> by Triangle Mid-segment TheoremBecause V is the midpoint of XY and T is the midpoint of XZsoVT is parallel to YZwe have[tex]YZ=18\ yd[/tex]substitute [tex]VT=\frac{18}{2}=9\ yd[/tex]