Question

given: parallelogram abcd has vertices a(0,5), b(7,12), c(10,9), and d(3,2). prove: abcd is a rectangle. martin uses the diagonals of the parallelogram to complete the proof. which is a key statement in the proof? bd = √116 the slope of bd is 5/2. ac = √84 the slope of ac is -2/5.

Explanation:

Step1: Recall rectangle - property

A parallelogram is a rectangle if and only if its diagonals are equal.

Step2: Use distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For diagonal $AC$ with $A(0,5)$ and $C(10,9)$:
$AC=\sqrt{(10 - 0)^2+(9 - 5)^2}=\sqrt{100 + 16}=\sqrt{116}$.
For diagonal $BD$ with $B(7,12)$ and $D(3,2)$:
$BD=\sqrt{(3 - 7)^2+(2 - 12)^2}=\sqrt{(-4)^2+(-10)^2}=\sqrt{16 + 100}=\sqrt{116}$.
Since $AC = BD=\sqrt{116}$, the key statement is $BD=\sqrt{116}$.

Answer:

$BD=\sqrt{116}$