Chapter 5.4, Problem 11CE

a.

To determine

To find: the common property of the red figures.

Answer to Problem 11CE

Opposite sides are equal and parallel in all the red figures.

Explanation of Solution

Given:

The new quadrilaterals are formed by joining the midpoints of the sides of the quadrilaterals.

  

The red figures are made by joining the midpoints of the sides of the quadrilaterals.

The common property of the formed quadrilaterals is that opposite sides are equal and parallel in all cases.

Conclusion:

Therefore, the common property is that the opposite sides are equal and parallel.

b.

To determine

To prove: that opposite sides are equal and parallel.

Explanation of Solution

Given:

The new quadrilaterals are formed by joining the midpoints of the sides of the quadrilaterals.

  

Take one figure from the above four figures.

  

Join $DB$ .

In $\Delta ABD$ , as $S$ and $P$ are midpoints of $AD$ and $AB$ respectively.

By Midpoint Theorem,

  $SP$ is parallel to $DB$ ….(i)

and $SP=\frac{1}{2}DB$ ….(ii).

In $\Delta CBD$ , as $R$ and $Q$ are midpoints of $CD$ and $CB$ respectively.

By Midpoint Theorem,

  $RQ$ is parallel to $DB$ ….(iii)

and $RQ=\frac{1}{2}DB$ ….(iv).

  $\therefore$ from equation (i) and equation (ii),

  $SP$ is parallel to $DB$ is parallel to $RQ$

  $\Rightarrow SP$ is parallel to $RQ$ .

From equation (ii) and equation (iv),

  $SP=\frac{1}{2}DB=RQ$

  $\Rightarrow SP=RQ$

Similarly by joining $AC$ ,

  $PC=SR$ and $PC$ is parallel to $SR$ .

Similarly the same can be proven for other figures.

Conclusion:

Therefore, the opposite sides of red figures are equal and parallel.