Question

the graph shows quadrilaterals pqrs and uvwx.
is pqrs similar to uvwx? justify your answer.

Explanation:

Step1: Find the coordinates of the vertices

Assume the coordinates:
Let \(U(-5,6)\), \(V(-1,3)\), \(W(-3,6)\), \(X(-6,9)\)
Let \(P(-3, - 5)\), \(Q(-1,-2)\), \(R(-5,-3)\), \(S(-9,-6)\)

Step2: Calculate the side - length ratios using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For quadrilateral \(UVWX\):
\(d_{UV}=\sqrt{(-1+5)^2+(3 - 6)^2}=\sqrt{16 + 9}=\sqrt{25}=5\)
\(d_{VW}=\sqrt{(-3 + 1)^2+(6 - 3)^2}=\sqrt{4+9}=\sqrt{13}\)
For quadrilateral \(PQRS\):
\(d_{PQ}=\sqrt{(-1 + 3)^2+(-2 + 5)^2}=\sqrt{4 + 9}=\sqrt{13}\)
\(d_{QR}=\sqrt{(-5+1)^2+(-3 + 2)^2}=\sqrt{16+1}=\sqrt{17}\)

Since the ratios of the corresponding side - lengths are not equal (e.g., \(\frac{d_{UV}}{d_{PQ}}=\frac{5}{\sqrt{13}}
eq1\) and \(\frac{d_{VW}}{d_{QR}}=\frac{\sqrt{13}}{\sqrt{17}}
eq1\)), and there is no center of dilation that can map one quadrilateral to the other such that all corresponding sides are in proportion.

Answer:

No, \(PQRS\) is not similar to \(UVWX\) because the ratios of their corresponding side - lengths are not equal.