GEOMETRY, 2008

 PAGE: 352 SET: Exercises PROBLEM: 6

Explore:   Plot the vertices on a coordinate plane.

Plan:

If the diagonals are perpendicular, then ABCD is either a rhombus or a square.

The diagonals of a rectangle are congruent. If the diagonals are congruent and perpendicular, then ABCD is a square.

Solve:

Use the Distance Formula to compare the lengths of the diagonals.

The distance formula is  .

(Substitute )

(Simplify)

(Substitute )

(Simplify)

Use slope to determine whether the diagonals are perpendicular.

The slope formula is .

Slope of          (Substitute )

Slope of             (Substitute )

Since the slope of is the negative reciprocal of the slope of , the diagonals are perpendicular. and   have the same measure, so the diagonals are congruent. ABCD is a rhombus, a rectangle, and a square. ABCD is a rhombus, a rectangle, and a square.

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