Question

the graph shows quadrilaterals hjk and rstu. is hjk similar to rstu? justify your answer.

Explanation:

Step1: Find the lengths of corresponding sides

• For \(HIJK\):
• Length of \(HI\): \(y\)-coordinate difference. \(H(6,3)\), \(I(6, - 6)\). \(|3-(-6)| = 9\)
• Length of \(JK\): \(x\)-coordinate difference. \(J(-3,-6)\), \(K(-3,6)\). \(| - 6 - 6|=12\)
• Length of \(HK\): Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). \(H(6,3)\), \(K(-3,6)\). \(d=\sqrt{(6 + 3)^2+(3 - 6)^2}=\sqrt{81 + 9}=\sqrt{90}=3\sqrt{10}\)
• For \(RSTU\):
• Length of \(RS\): \(y\)-coordinate difference. \(R(2,2)\), \(S(2,-2)\). \(|2-(-2)| = 4\)
• Length of \(TU\): \(x\)-coordinate difference. \(T(-1,-2)\), \(U(-1,3)\). \(| - 2-3| = 5\)
• Length of \(RU\): Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). \(R(2,2)\), \(U(-1,3)\). \(d=\sqrt{(2 + 1)^2+(2 - 3)^2}=\sqrt{9+1}=\sqrt{10}\)

Step2: Check the ratios of corresponding sides

• \(\frac{HI}{RS}=\frac{9}{4}\), \(\frac{JK}{TU}=\frac{12}{5}\), \(\frac{HK}{RU}=\frac{3\sqrt{10}}{\sqrt{10}} = 3\)
• Since \(\frac{9}{4}

eq\frac{12}{5}
eq3\), the ratios of corresponding sides are not equal.

Answer:

\(HIJK\) is not similar to \(RSTU\) because the ratios of their corresponding sides are not equal.