Quadrilaterals vertices PQRS

Question

Quadrilateral PQRS has vertices P(3,2), Q(0,4), R(-2,1) and S(1,-1)

a) Verify that quadrilateral PQRS is a square.

b) Find the perimeter of the rectangle.

Answer 1

a) There is one other way to show the quadrilateral is a square. That is to see if the lengths of the two diagonals agree.

Let the diagonals connect points Q and S and R and P.

The distance formula is sqrt[ (x1-x2)² + (y1-y2)² }

The distance between points Q and S is sqrt[ (0 - 1)² + (4 - (-1))² } = sqrt [1 + 25] = sqrt(26)

Similarly, the distance between points R and P is sqrt[ (-2 - 3)² + (1 - 2)² } = sqrt [25 + 1] = sqrt(26)

The diagonals agree, so the figure has four right angles.

The distances between points Q and P and P and S can be found to be sqrt(13) using the same methodology.

The combination of 4 right angles and equal lengths of sides determines the figure is a square.

b) By the Pythagorean theorem H² = A² + B². Here H is the diagonal that was just calculated. A = B because the giure is a square.

H² = 2A^2. 26 = 2A². 13 = A². Thus A = B = sqrt(13)

The perimeter of the square is 4*sqrt(13).

Answer 2

1. Verify that the sides are all equal, using the distance formula.
2. Verify that the corners are all right angles. The easiest way might be to use the reciprocal-slopes formula: m₁*m₂ = -1

Answer 3

The distance between two points can be found using the formula sqrt( (x₂-x₁)²+(y₂-y₁)² )

The distance between P and Q is sqrt( (0-3)²+(4-2)²)=sqrt(9+4)=sqrt(13)

The distance between Q and R is sqrt( (-2-0)²+(1-4)² )=sqrt(4+9)=sqrt(13)

Continue with the other points to find they all equal sqrt(13), so the shape is in fact a square and the perimeter is 4sqrt(13).