Question

question 8 of 39
martin wants to use coordinate geometry to prove that the opposite sides of
a rectangle are congruent. he places parallelogram abcd in the coordinate
plane so that a is (0,0), b is (a,0), c is (a,b), and d is (0,b).
what formula can he use to determine the distance from point d to point a?
a. $\sqrt{(0 - 0)^2+(b - 0)^2}=\sqrt{b^2}=b$
b. $\sqrt{(a - a)^2+(b - 0)^2}=\sqrt{b^2}=b$
c. $(0 - 0)^2+(b - 0)^2=b^2$
d. $(a - a)^2+(b - 0)^2=b^2$
submit

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Identify the coordinates of points \(A\) and \(D\)

Point \(A=(0,0)\) (where \(x_1 = 0,y_1 = 0\)) and point \(D=(0,b)\) (where \(x_2 = 0,y_2 = b\)).

Step3: Substitute into the distance formula

Substitute \(x_1 = 0,y_1 = 0,x_2 = 0,y_2 = b\) into \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we get \(d=\sqrt{(0 - 0)^2+(b - 0)^2}=\sqrt{0 + b^2}=\sqrt{b^2}=b\).

Option B uses incorrect \(x\)-coordinates (should not involve \(a\) for points \(A\) and \(D\)). Options C and D do not apply the distance formula (missing the square - root).

Answer:

A. $\sqrt{(0 - 0)^2+(b - 0)^2}=\sqrt{b^2}=b$