Question

kite bcde is shown on the graph below. bcde is dilated by a scale factor of $\frac{1}{2}$ centered at $(-8,3)$ to create $bcde$.
what is the perimeter of $bcde$?
write your answer as a whole number or as a decimal rounded to the nearest tenth.
units

Explanation:

Step1: Find the lengths of the sides of kite \( BCDE \)

Use the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).

• For \( BC \): \( B(-1,6) \), \( C(5,8) \)

\( BC=\sqrt{(5 + 1)^2+(8 - 6)^2}=\sqrt{36 + 4}=\sqrt{40}=2\sqrt{10}\)

• For \( CD \): \( C(5,8) \), \( D(6,2) \)

\( CD=\sqrt{(6 - 5)^2+(2 - 8)^2}=\sqrt{1+36}=\sqrt{37}\)

• For \( DE \): \( D(6,2) \), \( E(-2,-8) \)

\( DE=\sqrt{(-2 - 6)^2+(-8 - 2)^2}=\sqrt{64 + 100}=\sqrt{164}=2\sqrt{41}\)

• For \( EB \): \( E(-2,-8) \), \( B(-1,6) \)

\( EB=\sqrt{(-1 + 2)^2+(6 + 8)^2}=\sqrt{1+196}=\sqrt{197}\)

The perimeter of \( BCDE \) is \( P = 2\sqrt{10}+\sqrt{37}+2\sqrt{41}+\sqrt{197}\approx2\times3.16+6.08+2\times6.40+14.04\approx6.32+6.08+12.8+14.04 = 39.24\)

Step2: Apply the dilation

Since the scale factor \( k=\frac{1}{2} \), the perimeter of \( B'C'D'E' \) is \( P'=k\times P \)
\( P'=\frac{1}{2}\times39.24 = 19.62\approx19.6\)

Answer:

\( 19.6 \)