Question

pentagon abcde models a running track in a park. a section of track connects point b to point d. two runners are taking separate routes through the park: one will follow the path along the perimeter of rectangle abde, and the other will follow the track along the perimeter of triangle bcd. they decide to see who, if either, will run the farthest by finding the ratio of the perimeter of rectangle abde to the perimeter of triangle bcd. what is the ratio of the perimeter of rectangle abde to the perimeter of triangle bcd? 2 2/3 1

Explanation:

Step1: Find the perimeter of rectangle \(ABDE\)

The coordinates are \(A(0,6)\), \(B(2,6)\), \(D(2,0)\), \(E(0,0)\).
Length \(AB = 2-0=2\) units, length \(BD=6 - 0=6\) units.
Perimeter of rectangle \(P_{ABDE}=2\times(AB + BD)=2\times(2 + 6)=16\) units.

Step2: Find the lengths of sides of \(\triangle BCD\)

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(BC\): \(B(2,6)\), \(C(6,3)\), \(BC=\sqrt{(6 - 2)^2+(3 - 6)^2}=\sqrt{16 + 9}=\sqrt{25}=5\) units.
For \(CD\): \(C(6,3)\), \(D(2,0)\), \(CD=\sqrt{(2 - 6)^2+(0 - 3)^2}=\sqrt{16+9}=\sqrt{25}=5\) units.
For \(BD\): \(B(2,6)\), \(D(2,0)\), \(BD = 6\) units.
Perimeter of \(\triangle BCD\), \(P_{BCD}=5 + 5+6=16\) units.

Step3: Calculate the ratio

Ratio \(=\frac{P_{ABDE}}{P_{BCD}}=\frac{16}{16}=1\)

Answer:

\(1\)