Question

the graph shows quadrilaterals stuv and cdef.

Explanation:

Step1: Calculate the ratios of corresponding sides

• For side \(ST\) and \(CD\):
• The length of \(ST\) can be calculated using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(S(3,0)\) and \(T(2,2)\), \(ST=\sqrt{(3 - 2)^2+(0 - 2)^2}=\sqrt{1 + 4}=\sqrt{5}\).
• For \(C(9,0)\) and \(D(6,6)\), \(CD=\sqrt{(9 - 6)^2+(0 - 6)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}\). The ratio of \(ST\) to \(CD\) is \(\frac{\sqrt{5}}{3\sqrt{5}}=\frac{1}{3}\).
• For side \(TU\) and \(DE\):
• For \(T(2,2)\) and \(U(- 2,2)\), \(TU=\vert2-(-2)\vert = 4\).
• For \(D(6,6)\) and \(E(-6,6)\), \(DE=\vert6-(-6)\vert = 12\). The ratio of \(TU\) to \(DE\) is \(\frac{4}{12}=\frac{1}{3}\).
• For side \(UV\) and \(EF\):
• For \(U(-2,2)\) and \(V(-3,0)\), \(UV=\sqrt{(-2 + 3)^2+(2 - 0)^2}=\sqrt{1+4}=\sqrt{5}\).
• For \(E(-6,6)\) and \(F(-9,0)\), \(EF=\sqrt{(-6 + 9)^2+(6 - 0)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}\). The ratio of \(UV\) to \(EF\) is \(\frac{\sqrt{5}}{3\sqrt{5}}=\frac{1}{3}\).
• For side \(VS\) and \(FC\):
• For \(V(-3,0)\) and \(S(3,0)\), \(VS=\vert-3 - 3\vert=6\).
• For \(F(-9,0)\) and \(C(9,0)\), \(FC=\vert-9 - 9\vert = 18\). The ratio of \(VS\) to \(FC\) is \(\frac{6}{18}=\frac{1}{3}\).

Step2: Check the angles

Since the sides of \(STUV\) and \(CDEF\) are parallel (both have horizontal sides \(TU\), \(DE\), \(VS\), \(FC\) and non - horizontal sides with the same slope). Corresponding angles are equal.

Answer:

Yes, \(STUV\) is similar to \(CDEF\) because all corresponding sides are in proportion (\(\frac{1}{3}\)) and corresponding angles are equal.