Question

1. Prove that quadrilateral DOGS is

Answer 1

Answer:

DOGS is a parallelogram.

Step-by-step explanation:

Given the quadrilateral DOGS with coordinates D(1, 1), O(2, 4),  G(5, 6), and S(4,3).

To prove that it is a parallelogram, we need to show that the opposite lengths are equal. That is:

• |DO|=|GS|
• |OG|=|SD|

Using the Distance Formula

$Distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

For D(1, 1) and O(2, 4)

$|DO|=\sqrt{(2-1)^2+(4-1)^2}=\sqrt{1^2+3^2}=\sqrt{10} \:Units$

For G(5, 6), and S(4,3).

$|GS|=\sqrt{(4-5)^2+(3-6)^2}=\sqrt{(-1)^2+(-3)^2}=\sqrt{10}\:Units$

For O(2, 4) and G(5, 6)

$|OG|=\sqrt{(5-2)^2+(6-4)^2}=\sqrt{(3)^2+(2)^2}=\sqrt{13}\:Units$

For S(4,3) and D(1, 1)

$|SD|=\sqrt{(1-4)^2+(1-3)^2}=\sqrt{(-3)^2+(-2)^2}=\sqrt{13}\:Units$

Since:

• |DO|=|GS|
• |OG|=|SD|

Then, quadrilateral DOGS is a parallelogram.