Question

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

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Explanation:

Step1: Find length of AE

Use distance formula for A(0,6) and E(11,10):
$\text{AE} = \sqrt{(11-0)^2 + (10-6)^2} = \sqrt{121 + 16} = \sqrt{137}$

Step2: Find length of AF

Use distance formula for A(0,6) and F(5,2):
$\text{AF} = \sqrt{(5-0)^2 + (2-6)^2} = \sqrt{25 + 16} = \sqrt{41}$

Step3: Find length of ED

Use distance formula for E(11,10) and D(14,14):
$\text{ED} = \sqrt{(14-11)^2 + (14-10)^2} = \sqrt{9 + 16} = 5$

Step4: Calculate area of ABEF

Area = $\text{AE} \times \text{AF} = \sqrt{137} \times \sqrt{41} = \sqrt{5617} \approx 74.95$

Step5: Calculate area of ACDF

Area of ACDF = Area of ABEF + Area of BCDE. First find area of BCDE: $\text{AE} \times \text{ED} = \sqrt{137} \times 5 \approx 58.58$. So Area of ACDF $\approx 74.95 + 58.58 = 133.53$

Step6: Find area ratio ABEF:ACDF

$\text{Ratio} = \frac{74.95}{133.53} \approx \frac{2}{3}$

Step7: Calculate perimeter of BCDE

Perimeter = $2(\text{AE} + \text{ED}) = 2(\sqrt{137} + 5) \approx 2(11.70 + 5) = 33.40$

Answer:

1. The ratio of the area of rectangle ABEF to the area of rectangle ACDF is $\boldsymbol{2:3}$
2. The area of rectangle ABEF is $\boldsymbol{\approx 74.95}$ square units
3. The perimeter of rectangle BCDE is $\boldsymbol{\approx 33.40}$ units