Question

rectangle abcd has the following vertices:
· a(-1,9)
· b(9,4)
· c(4,-6)
· d(-6,-1)
is rectangle abcd a square, and why?
choose 1 answer:
a yes, because ab = bc = cd = ad, and abcd is a rectangle.
b yes, because ab || cd and bc || ad, and abcd is a rectangle.
c no, because ab is longer than bc.
d no, because ab is not parallel to bc.

Explanation:

Step1: Calculate the length of \(AB\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(-1,9)\) and \(B(9,4)\), \(AB=\sqrt{(9 + 1)^2+(4 - 9)^2}=\sqrt{100 + 25}=\sqrt{125}=5\sqrt{5}\)

Step2: Calculate the length of \(BC\)

For \(B(9,4)\) and \(C(4,-6)\), \(BC=\sqrt{(4 - 9)^2+(-6 - 4)^2}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}\)

Step3: Calculate the length of \(CD\)

For \(C(4,-6)\) and \(D(-6,-1)\), \(CD=\sqrt{(-6 - 4)^2+(-1 + 6)^2}=\sqrt{100 + 25}=\sqrt{125}=5\sqrt{5}\)

Step4: Calculate the length of \(AD\)

For \(A(-1,9)\) and \(D(-6,-1)\), \(AD=\sqrt{(-6 + 1)^2+(-1 - 9)^2}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}\)

Since \(AB = BC=CD = AD\) and \(ABCD\) is a rectangle (given). A square is a special - case of a rectangle where all sides are equal.

Answer:

A. Yes, because \(AB = BC = CD = AD\), and \(ABCD\) is a rectangle.