Question

2. Given a quadrilateral with vertices (−1, 3), (1, 5), (5, 1), and (3,−1):

a. Prove that quadrilateral is a rectangle.

Answer 1

Step-by-step explanation:

In every rectangle, the two diagonals have the same length. If a quadrilateral's diagonals have the same length, that doesn't mean it has to be a rectangle, but if a parallelogram's diagonals have the same length, then it's definitely a rectangle.

So first of all, let's prove this is a parallelogram. The basic definition of a parallelogram is that it is a quadrilateral where both pairs of opposite sides are parallel.

So let's name the vertices as:

`A(-1,3) \\ \\ B(1,5) \\ \\ C(5,1) \\ \\ D(3,-1)`

First pair of opposite sides:

Slope:

`\text{For AB}: \\ \\ m=(5-3)/(1-(-1))=1 \\ \\ \\ \text{For CD}: \\ \\ m=(1-(-1))/(5-3)=1 \\ \\ \\ \text{So AB and CD are parallel}`

Second pair of opposite sides:

Slope:

`\text{For BC}: \\ \\ m=(1-5)/(5-1)=-1 \\ \\ \\ \text{For AD}: \\ \\ m=(-1-3)/(3-(-1))=-1 \\ \\ \\ \text{So BC and AD are parallel}`

So in fact this is a parallelogram. The other thing we need to prove is that the diagonals measure the same. Using distance formula:

`d=\sqrt{(y_(2)-y_(1))^2+(x_(2)-x_(1))^2} \\ \\ \\ Diagonal \ BD: \\ \\ d=√((5-(-1))^2+(1-3)^2)=2√(10) \\ \\ \\ Diagonal \ AC: \\ \\ d=√((3-1)^2+(-5-1)^2)=2√(10) \\ \\ \\`

So the diagonals measure the same, therefore this is a rectangle.