Question

Quadrilateral NORA has vertices N(3,2), (7,0), R(11,2),

and A(7,4). Use coordinate geometry to prove that
a) quadrilateral NORA is a rhombus, and
b) quadrilateral NORA is not a square.

Answer 1

By calculating side lengths and checking for perpendicularity using the distance and slope formulas, we prove that quadrilateral NORA is a rhombus but not a square.

Step-by-step explanation:

Proving that quadrilateral NORA is a rhombus requires showing that all sides are of equal length. We calculate this using the distance formula d = √((x_2-x_1)^2 + (y_2-y_1)^2). We find that sides NO, OR, RA, and AN are all equal, satisfying the condition for NORA to be a rhombus.

Proving that NORA is not a square involves showing that while all sides are equal, the angles are not all 90 degrees. Using the slope formula m = (y_2-y_1) / (x_2-x_1), we check for perpendicularity between adjacent sides. Since the product of slopes of NO and OR is not equal to -1, these sides are not perpendicular, confirming NORA is not a square.