Chapter 11.3, Problem 18PSC

To determine

To Calculate:The formula to find the length of PQ

Explanation of Solution

Given information : Trapezoid ABCD , in which X and Y are midpoint of sides and P and Q are midpoint of diagonals

Formula used : Midpoint theorem

Calculation: Draw a line AG passing from P to DC .

Now, in $\Delta DPG$ and $\Delta APB$ ,

  $\angle DPG=\angle APB$ (Vertically Opposite Angles)

  $\angle PDG=\angle ABP$ (Alternate Interior Angles)

  $DP=PB$ (P is the midpoint of BD)

  $\Rightarrow \Delta DPG\cong \Delta APB$ (By SAS Congruency Rule)

  $\therefore DG=AB$ and $AP=PG$ (By Corresponding Parts of Congruent Triangles)

Now, in $\Delta AGC$ ,

P is midpoint of AG and AC,

  $\therefore PQ\parallel CG$ and $PQ=\frac{1}{2}(CG)$ (By midpoint theorem)

  $\Rightarrow PQ=\frac{1}{2}(DC-DG)$