AREAS OF QUADRILATERALS

WITH PERPENDICULAR DIAGONALS

DEFINITIONS

SQUARE: A square is a quadrilateral with four congruent sides

and four right angles.

RHOMBUS: A rhombus is a quadrilateral with four congruent sides.

KITE: A kite is a quadrilateral with two pairs of adjacent sides

congruent and no opposite sides congruent.

TRAPEZOID: A trapezoid is a quadrilateral with exactly one pair

of parallel sides.

CASE 1: QUADRILATERALS WHOSE DIAGONALS ARE

CONGRUENT PERPENDICULAR BISECTORS

OF ONE ANOTHER

1.  Click on “Graph” and “Create Axes”.

2.  Plot the points (1,0), (0,1), (-1,0),and (0,-1), and connect them, in this order, to form a quadrilateral.

(a)  Click on the “Point Tool”.

(b)  Hold the “Shift” key down while you place points

at each of the locations indicated above.

(c)  Click on the “Arrow”.

(d)  Unselect the points.

(e)  Select points (1,0) and (0,1) while holding “Shift”.

(f)  Click on “Construct”, then “Segment”.

(g)  Unselect.

(h)  Select points (0,1) and (-1,0).

(i)  Click on “Construct”, then “Segment”.

(j)  Unselect.

(k)  Select points (-1,0) and (0,-1).

(l)  Click on “Construct”, then “Segment”.

(m)  Unselect.

(n)  Select points (0,-1) and (1,0).

(o)  Click on “Construct”, then “Segment”.

(p)  Unselect.

3.  Verify that the quadrilateral formed is a square.

[Check to see that all sides have the same length.]

(a)  Select the two endpoints of one side.

(b)  Click on “Measure”, then “Distance”.

(c)  Continue this procedure for all four sides.

[Check to see that all angles are right angles.]

(a)  Select three vertices.

(b)  Click on “Measure”, then “Angle”.

(c)  Continue this procedure to verify that the measure of each angle is 90o.

4.  Are the diagonals congruent perpendicular bisectors of one another?

5.  Record the length of each diagonal in the chart below.

6.  Use Geometer’s Sketchpad to determine the area of the square.

(a)  Select the vertices of the square.

(b)  Click on “Construct”, then “Polygon Interior”.

(c)  Click on “Measure”, then “Area”.

(d)  Write the area in the chart.

7.  Continue the procedure with the vertices of other squares.

8.  What relationship do you find between the lengths of the diagonals of each square and its area?

VERTICES / HORIZONTAL
DIAGONAL / VERTICAL
DIAGONAL / AREA
(1,0) (0,1) (-1,0) (0,-1)
(1½,0) (0,1½) (-1½,0) (0,-1½)
(2,0) (0,2) (-2,0) (0,-2)
(2½,0) (0,2½) (-2½,0) (0,-2½)
(3,0) (0,3) (-3,0) (0,-3)
(3½,0) (0,3½) (-3½,0) (0,-3½)
(4,0) (0,4) (-4,0) (0,-4)
(4½,0) (0,4½) (-4½,0) (0,-4½)

CASE 2: QUADRILATERALS WHOSE DIAGONALS ARE

PERPENDICULAR BISECTORS OF ONE ANOTHER

BUT ARE NOT CONGRUENT

1.  Plot the points (2,0), (0,1), (-2,0), and (0,-1), and connect them, in this

order, to form a quadrilateral.

2.  Verify that this quadrilateral is a rhombus.

[Check to see that all sides have the same length.]

3.  Are the diagonals perpendicular bisectors of one another?

4.  Are the diagonals congruent?

5.  Record the length of each diagonal in the chart below.

6.  Use Geometer’s Sketchpad to determine the area of the rhombus and record it in the chart.

7.  Continue the procedure with the vertices of other rhombi.

(Write the coordinates of the vertices in the chart.)

8.  What relationship do you find between the lengths of the diagonals of

each rhombus and its area?

VERTICES / HORIZONTAL
DIAGONAL / VERTICAL
DIAGONAL / AREA
(2,0) (0,1) (-2,0) (0,-1)

CASE 3: QUADRILATERALS IN WHICH ONLY ONE

DIAGONAL IS A PERPENDICULAR BISECTOR

OF THE OTHER

1.  Plot the points (1,0), (0,2), (-1,0), and (0,-3), and connect them, in this

order, to form a quadrilateral.

2.  Verify that this quadrilateral is a kite.

[Check to see that two pairs of adjacent sides have the same length,

but that opposite sides do not have the same length.]

3.  Are the diagonals perpendicular?

4.  Do the diagonals bisect one another?

5.  Record the length of each diagonal in the chart below.

6.  Use Geometer’s Sketchpad to determine the area of the kite and record it in the chart.

7.  Continue the procedure with the vertices of other kites.

(Write the coordinates of the vertices in the chart.)

8.  What relationship do you find between the lengths of the diagonals of

each kite and its area?

VERTICES / HORIZONTAL
DIAGONAL / VERTICAL
DIAGONAL / AREA
(1,0) (0,2) (-1,0) (0,-3)

CASE 4: QUADRILATERALS WHOSE DIAGONALS ARE

PERPENDICULAR, BUT DO NOT BISECT EACH

OTHER

1.  Plot the points (2,0), (0,1), (-1,0), and (0,-2), and connect them, in this

order, to form a quadrilateral.

2.  Are the diagonals perpendicular?

3.  Do either of the diagonals bisect the other?

4.  Record the length of each diagonal in the chart below.

5.  Use Geometer’s Sketchpad to determine the area of the quadrilateral and record it in the chart.

6.  Continue the procedure with the vertices of other such quadrilaterals.

(Write the coordinates of the vertices in the chart.)

7.  What relationship do you find between the lengths of the diagonals of

each of these quadrilaterals and its area?

VERTICES / HORIZONTAL
DIAGONAL / VERTICAL
DIAGONAL / AREA
(2,0) (0,1) (-1,0) (0,-2)
(3,0) (0,1) (-1,0) (0,-2)
THEOREM

The area of a quadrilateral

whose diagonals are perpendicular to one another

is one-half the product of those diagonals.

Write a convincing argument verifying the truth of this theorem.

GIVEN: Quadrilateral ABCD with

PROVE: Area of quadrilateral ABCD = ½ AC . BD

What kind of reasoning did you use to discover this new area formula?

Circle One: INDUCTIVE DEDUCTIVE

What kind of reasoning did you use to prove that the formula worked?

Circle One: INDUCTIVE DEDUCTIVE

RELATED INVESTIGATIONS

1.  Which quadrilaterals in your chart for Case 4 are trapezoids?

[Verify that the quadrilateral is a trapezoid by checking the slopes of

the sides. Exactly one pair of sides must be parallel.]

(a)  Select a side.

(b)  Click on “Measure”, then “Slope”.

(c)  Unselect.

(d)  Continue this procedure to determine the slopes

of all four sides.

2.  Construct quadrilaterals whose vertices do not lie on either the x-axis

or the y-axis. Determine the lengths of the diagonals and the area of

each quadrilateral to demonstrate the truth of the theorem.

3.  Is this theorem true for concave, as well as convex, quadrilaterals whose diagonals are perpendicular?

RELATED WEB SITES

http://www.geom.umn.edu/docs/reference/CRC-formulas/node23.html

http://www.tenet.edu/teks/math/clarifying/clgeoquads.html

http://www.math.okstate.edu/~rpsc/Ideas/DiagQuad.html

http://www.google.com/search?q=Diagonals+and+quadrilaterals

THEOREM

GIVEN: Quadrilateral ABCD with

PROVE: Area of quadrilateral ABCD = ½ AC . BD

PROOF:

Area of ABC = ½ AC . BE

Formula for Area of a Triangle

Area of ADC = ½ AC . ED

Area of quadrilateral ABCD

= Area of ABC + Area of ADC The area of a region is the sum

of the areas of its non-overlapping parts.

= ½ AC . BE + ½ AC . ED Substitution Property

= ½ AC (BE + ED) Distributive Property

= ½ AC . BD Definition of Between

INSTRUCTIONAL GOALS AND OBJECTIVES

WV IGO G.16

Develop and apply formulas for area, perimeter, surface area, and volume and apply them in the modeling of practical problems.

WV IGO G.22

Using the Cartesian coordinate system, find the dimensions of a polygon, given the coordinates of the polygon.

WV IGO G.8

Explore and identify properties of quadrilaterals and verify properties for parallelogram, rectangle, rhombus, square, and trapezoid.

WV IGO G.23

Use appropriate software to practice and master geometry and applied geometry instructional objectives.

WV IGO G.24

Use a calculator to perform operations on whole numbers, fractions, and decimals.

WV IGO G.4

Construct logical arguments using various formats with emphasis on paragraph form, flow proofs, and indirect approaches.

WV IGO G.2

Differentiate between inductive and deductive reasoning.

Karen Yoho