Question

in the figure, the ratio of the area of rectangle \\(abef\\) to the area of rectangle \\(acdf\\) is choose an answer.
if the coordinates of point \\(a\\) are \\((0,6)\\), the area of rectangle \\(abef\\) is choose an answer square units, and the area of rectangle \\(acdf\\) is choose an answer square units.

Explanation:

Find the dimensions of rectangle ABEF

We are given the coordinates of point \(A(0,6)\), point \(F(5,2)\), and point \(E(11,10)\).
First, we calculate the length of the side \(AF\) using the distance formula:

$$ AF = \sqrt{(5 - 0)^2 + (2 - 6)^2} = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} $$

Next, we calculate the length of the side \(FE\) using the distance formula:

$$ FE = \sqrt{(11 - 5)^2 + (10 - 2)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $$

Since \(ABEF\) is a rectangle, its area is:

$$ \text{Area}(ABEF) = AF \times FE = \sqrt{41} \times 10 = 10\sqrt{41} $$

Find the dimensions of rectangle ACDF

The rectangle \(ACDF\) shares the side \(AF\) with rectangle \(ABEF\). Thus, the width of \(ACDF\) is \(AF = \sqrt{41}\).
The length of \(ACDF\) is \(FD\). We calculate the distance \(FD\) using the coordinates of \(F(5,2)\) and \(D(14,14)\):

$$ FD = \sqrt{(14 - 5)^2 + (14 - 2)^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 $$

The area of rectangle \(ACDF\) is:

$$ \text{Area}(ACDF) = AF \times FD = \sqrt{41} \times 15 = 15\sqrt{41} $$

Calculate the ratio of the areas

We find the ratio of the area of rectangle \(ABEF\) to the area of rectangle \(ACDF\):

$$ \text{Ratio} = \frac{\text{Area}(ABEF)}{\text{Area}(ACDF)} = \frac{10\sqrt{41}}{15\sqrt{41}} = \frac{10}{15} = \frac{2}{3} $$

Alternatively, since both rectangles share the same width \(AF\), the ratio of their areas is simply the ratio of their lengths:

$$ \frac{\text{Area}(ABEF)}{\text{Area}(ACDF)} = \frac{FE}{FD} = \frac{10}{15} = \frac{2}{3} $$

Answer:

• The ratio of the area of rectangle \(ABEF\) to the area of rectangle \(ACDF\) is \(2:3\) (or \(\frac{2}{3}\)).
• If the coordinates of point \(A\) are \((0,6)\), the area of rectangle \(ABEF\) is \(10\sqrt{41}\) square units.
• The area of rectangle \(ACDF\) is \(15\sqrt{41}\) square units.