Question

the graph shows quadrilaterals hijk and nopq. is hijk congruent to nopq? justify your answer.

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For quadrilateral \(NOPQ\):

• \(NO\): \(N(3,4)\), \(O(4,8)\)

\(d_{NO}=\sqrt{(4 - 3)^2+(8 - 4)^2}=\sqrt{1 + 16}=\sqrt{17}\)

• \(OP\): \(O(4,8)\), \(P(- 2,7)\)

\(d_{OP}=\sqrt{(-2 - 4)^2+(7 - 8)^2}=\sqrt{36+1}=\sqrt{37}\)

• \(PQ\): \(P(-2,7)\), \(Q(-3,3)\)

\(d_{PQ}=\sqrt{(-3 + 2)^2+(3 - 7)^2}=\sqrt{1 + 16}=\sqrt{17}\)

• \(QN\): \(Q(-3,3)\), \(N(3,4)\)

\(d_{QN}=\sqrt{(3 + 3)^2+(4 - 3)^2}=\sqrt{36 + 1}=\sqrt{37}\)

For quadrilateral \(HIJK\):

• \(HI\): \(H(-1,-4)\), \(I(-2,-8)\)

\(d_{HI}=\sqrt{(-2 + 1)^2+(-8 + 4)^2}=\sqrt{1+16}=\sqrt{17}\)

• \(IJ\): \(I(-2,-8)\), \(J(6,-7)\)

\(d_{IJ}=\sqrt{(6 + 2)^2+(-7 + 8)^2}=\sqrt{64 + 1}=\sqrt{65}\)

• \(JK\): \(J(6,-7)\), \(K(4,-3)\)

\(d_{JK}=\sqrt{(4 - 6)^2+(-3 + 7)^2}=\sqrt{4 + 16}=\sqrt{20}\)

• \(KH\): \(K(4,-3)\), \(H(-1,-4)\)

\(d_{KH}=\sqrt{(-1 - 4)^2+(-4 + 3)^2}=\sqrt{25+1}=\sqrt{26}\)

Step2: Compare side - lengths

Since the side - lengths of \(NOPQ\) (\(\sqrt{17},\sqrt{37},\sqrt{17},\sqrt{37}\)) and \(HIJK\) (\(\sqrt{17},\sqrt{65},\sqrt{20},\sqrt{26}\)) are not equal.

Answer:

No, \(HIJK\) is not congruent to \(NOPQ\) because their corresponding side - lengths are not equal.