Chapter 11.4, Problem 13PSC

a

To determine

To compare : The area of quadrilaterals ABCD and XQYP.

Answer to Problem 13PSC

The areas are conjectured to be congruent

Explanation of Solution

Given information : The following information has been given

X is the midpoint of AB

Y is the midpoint of CD

  $PD=QC$

We draw the lines XY and PQ . We then name their intersection point as O .

Now, we can break the given figures into eight different triangles and see the congruency of the figures.

Also, congruent figures have congruent areas.

b

To determine

To prove : The conjecture that area of $ABCD\cong XQYP$

Explanation of Solution

Given information : The following information has been given

X is the midpoint of AB

Y is the midpoint of CD

  $PD=QC$

Formula used : A diagonal cuts a rectangle into two congruent triangles

Proof : We draw the lines XY and PQ . We then name their intersection point as O.

Now, we can break the given figures into four different rectangles.

In rectangle AXOP , we know that XP is the diagonal. Hence, we can say that

Area of $\Delta AXP\cong \Delta OPX$

In rectangle XBQO , we know that XQ is the diagonal. Hence, we can say that

Area of $\Delta XOQ\cong \Delta BQX$

In rectangle OQCY , we know that OY is the diagonal. Hence, we can say that

Area of $\Delta OCY\cong \Delta CQY$

In rectangle POYD , we know that YP is the diagonal. Hence, we can say that

Area of $\Delta OPY\cong \Delta DPY$

Now, we add up all the triangles, and we get

Area of $ABCD\cong XQYP$