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Given quadrilateral PQRS with vertices:
P(-4, 1), Q(0, 4), R(3, 0), and S(-1, -3).

4.1 Prove that PQRS is a rhombus.
4.2 Is PQRS a square? Justify your answer.

Answer :

PQRS is a rhombus because all four sides have equal length (5 units). Since its diagonals are equal and perpendicular to each other, PQRS is also a square.

Given quadrilateral PQRS with P(-4,1), Q(0,4), R(3,0), and S(-1,-3), let's prove that PQRS is a rhombus and determine if it is a square.

4.1 Prove that PQRS is a Rhombus

A rhombus is a quadrilateral where all four sides have equal length. To prove PQRS is a rhombus, we'll calculate the lengths of all four sides using the distance formula:

Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Length of PQ:
PQ = sqrt((0 - (-4))^2 + (4 - 1)^2) = sqrt(4^2 + 3^2) = sqrt(16 + 9) = 5

Length of QR:
QR = sqrt((3 - 0)^2 + (0 - 4)^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = 5

Length of RS:
RS = sqrt((3 - (-1))^2 + (0 - (-3))^2) = sqrt(4^2 + 3^2) = sqrt(16 + 9) = 5

Length of SP:
SP = sqrt((-1 - (-4))^2 + (-3 - 1)^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = 5

Since PQ = QR = RS = SP = 5, all four sides are equal, so PQRS is a rhombus.

4.2 Is PQRS a Square.

A square is a rhombus with all right angles. To determine if PQRS is a square, we need to check if the diagonals are equal and perpendicular.

Diagonal PR: sqrt((3 - (-4))^2 + (0 - 1)^2) = sqrt(7^2 + (-1)^2) = sqrt(49 + 1) = sqrt(50) = 5*sqrt(2)

Diagonal QS: sqrt((0 - (-1))^2 + (4 - (-3))^2) = sqrt(1^2 + 7^2) = sqrt(1 + 49) = sqrt(50) = 5*sqrt(2)

Both diagonals are equal. Now, we check their slopes:

Slope of PR = (0 - 1) / (3 - (-4)) = -1/7

Slope of QS = (4 - (-3)) / (0 - (-1)) = 7/1 = 7

The product of their slopes is -1/7 * 7 = -1, indicating that they are perpendicular. Since all sides are equal and diagonals are perpendicular, PQRS is both a rhombus and a square.

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