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Facts about Quadrilaterals Help (page 2)

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

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.

Quadrilaterals Facts about Quadrilaterals Rhombus Diagonals

Fig. 3-12 . The diagonals of a rhombus 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 .

Quadrilaterals Facts about Quadrilaterals Trapezoid Within Triangle

Fig. 3-13 . A trapezoid is formed by “chopping off” the top of a triangle.

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.

Quadrilaterals Facts about Quadrilaterals Median Of A Trapezoid

Fig. 3-14 . The median of a trapezoid.

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.

Quadrilaterals Facts about Quadrilaterals Median Of A Trapezoid

Fig. 3-14 . The median of a trapezoid, also showing a transversal line.

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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