Question

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in the figure, the ratio of the perimeter of rectangle abde to the perimeter of triangle bcd is. the area of polygon abcde is square units.

Explanation:

Step1: Find coordinates

Coordinates: \(A(0,6)\), \(B(2,6)\), \(D(2,0)\), \(E(0,0)\), \(C(6,3)\)

Step2: Calculate rectangle \(ABDE\) perimeter

Length \(AB = 2 - 0=2\), width \(AE=6 - 0 = 6\). Perimeter \(P_{ABDE}=2\times(2 + 6)=16\)

Step3: Calculate triangle \(BCD\) side lengths

\(BD = 6-0 = 6\), \(CD=\sqrt{(6 - 2)^{2}+(3 - 0)^{2}}=\sqrt{16 + 9}=5\), \(BC=\sqrt{(6 - 2)^{2}+(3 - 6)^{2}}=\sqrt{16+9}=5\). Perimeter \(P_{BCD}=6 + 5+5 = 16\)
Ratio \(P_{ABDE}:P_{BCD}=16:16 = 1\)

Step4: Calculate area of \(ABDE\)

\(A_{ABDE}=2\times6 = 12\)

Step5: Calculate area of \(\triangle BCD\)

Base \(BD = 6\), height \(h=6 - 2=4\). \(A_{\triangle BCD}=\frac{1}{2}\times6\times4 = 12\)
Total area \(A = 12+12=24\)

Answer:

Ratio is \(1\), area is \(24\) square units.