**Rhombus Diagonals**

Suppose we have a parallelogram defined by four points *P, Q, R* , and *S* . Let *D* be the diagonal connecting *P* and *R* ; let *E* be the diagonal connecting *Q* and *S* . If *D* is perpendicular to *E* , then the parallelogram is a rhombus (Fig. 3-12). The converse is also true: If a parallelogram is a rhombus, then *D* is perpendicular to *E* . A parallelogram is a rhombus if and only if its diagonals are perpendicular.

**Trapezoid Within Triangle**

Suppose we have a triangle defined by three points *P, Q* , and *R* . Let *S* be the midpoint of side *PR* , and let *T* be the midpoint of side *PQ* . Then line segments *ST* and *RQ* are parallel, and the figure defined by *STQR* is a trapezoid (Fig. 3-13). In addition, the length of line segment *ST* is half the length of line segment *RQ* .

**Median Of A Trapezoid**

Suppose we have a trapezoid defined by four points *P, Q, R* , and *S* . Let *T* be the midpoint of side *PS* , and let *U* be the midpoint of side *QR*. Line segment *TU* is called the *median* of trapezoid *PQRS*. The median of a trapezoid is always parallel to both the base and the top, and always splits the trapezoid into two other trapezoids. That is, polygons *PQUT* and *TURS* are both trapezoids (Fig. 3-14). In addition, the length of line segment *TU* is half the sum of the lengths of line segments *PQ* and *SR*. That is, the length of *TU* is equal to the average, or *arithmetic mean* , of the lengths of *PQ* and *SR*.

**Median With Transversal**

Look again at Fig. 3-14. Suppose *L* is a transversal line that crosses both the top of the large trapezoid (line segment *PQ* ) and the bottom (line segment *SR* ). Then *L* also crosses the median, line segment *TU* . Let *A* be the point at which *L* crosses *PQ* , let *B* be the point at which *L* crosses *TU* , and let *C* be the point at which *L* crosses *SR* . Then the lengths of line segments *AB* and *BC* are equal.

There is a second fact that should also be mentioned. Again, refer to Fig. 3-14. Suppose *PQRS* is a trapezoid, with sides *PQ* and *RS* parallel. Suppose *TU* is a line segment parallel to both *PQ* and *RS* , and that intersects both of the non-parallel sides of the trapezoid, that is, sides *PS* and *QR* . Let *L* be a transversal line that crosses all three parallel line segments *PQ, TU* , and *RS* , at the points *A, B* , and *C* respectively, as shown. In this scenario, if line segments *AB* and *BC* are equally long, then line segment *TU* is the median of the large trapezoid *PQRS*.

**Facts about Quadrilaterals Practice Problems**

**PROBLEM 1**

Suppose a particular plane figure has diagonals that are the same length, and in addition, they intersect at right angles. What can be said about this polygon?

**SOLUTION 1**

From the above rules, this polygon must be a rectangle, because its diagonals are the same length. But it must also be a rhombus, because its diagonals are perpendicular to each other. There’s only one type of polygon that can be both a rectangle and a rhombus, and that is a square. A square is a rhombus in which both pairs of opposite interior angles happen to have the same measure. A square is also a rectangle in which both pairs of opposite sides happen to be equally long.

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