Question

the graph shows quadrilaterals defg and pqrs.
is defg congruent to pqrs? justify your answer.

Explanation:

Step1: Find coordinates of vertices

For \(DEFG\): \(D(2,6)\), \(E(4,8)\), \(F(7,6)\), \(G(4,4)\).
For \(PQRS\): \(Q(-9,4)\), \(R(-6,7)\), \(S(-4,4)\), \(P(-6,2)\).

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

- \(DE\): \(\sqrt{(4 - 2)^2+(8 - 6)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\)
- \(EF\): \(\sqrt{(7 - 4)^2+(6 - 8)^2}=\sqrt{9 + 4}=\sqrt{13}\) (Wait, correction: \(E(4,8)\), \(F(7,6)\): \(\sqrt{(7 - 4)^2+(6 - 8)^2}=\sqrt{9 + 4}=\sqrt{13}\)? No, wait \(D(2,6)\), \(E(4,8)\): \(\sqrt{(4 - 2)^2+(8 - 6)^2}=\sqrt{4 + 4}=2\sqrt{2}\). \(E(4,8)\), \(F(7,6)\): \(\sqrt{(7 - 4)^2+(6 - 8)^2}=\sqrt{9 + 4}=\sqrt{13}\)? No, actually, let's recalculate \(DE\): \(x\)-difference \(2\), \(y\)-difference \(2\), so \(DE = \sqrt{2^2 + 2^2}=2\sqrt{2}\). \(EF\): \(x\)-difference \(3\), \(y\)-difference \(-2\), so \(\sqrt{3^2 + (-2)^2}=\sqrt{13}\)? Wait, no, \(D(2,6)\), \(G(4,4)\): \(x\)-difference \(2\), \(y\)-difference \(-2\), so \(DG=\sqrt{2^2 + (-2)^2}=2\sqrt{2}\). \(E(4,8)\), \(G(4,4)\): \(x\)-difference \(0\), \(y\)-difference \(-4\), so \(EG = 4\). Wait, maybe better to check \(PQRS\): \(Q(-9,4)\), \(R(-6,7)\): \(x\)-difference \(3\), \(y\)-difference \(3\), so \(QR=\sqrt{3^2 + 3^2}=3\sqrt{2}\). \(R(-6,7)\), \(S(-4,4)\): \(x\)-difference \(2\), \(y\)-difference \(-3\), so \(\sqrt{4 + 9}=\sqrt{13}\)? No, this is confusing. Wait, actually, \(DEFG\): Let's list all sides:
\(DE\): between \((2,6)\) and \((4,8)\): \(\sqrt{(4 - 2)^2 + (8 - 6)^2}=\sqrt{4 + 4}=2\sqrt{2}\)
\(EF\): between \((4,8)\) and \((7,6)\): \(\sqrt{(7 - 4)^2 + (6 - 8)^2}=\sqrt{9 + 4}=\sqrt{13}\)
\(FG\): between \((7,6)\) and \((4,4)\): \(\sqrt{(4 - 7)^2 + (4 - 6)^2}=\sqrt{9 + 4}=\sqrt{13}\)
\(GD\): between \((4,4)\) and \((2,6)\): \(\sqrt{(2 - 4)^2 + (6 - 4)^2}=\sqrt{4 + 4}=2\sqrt{2}\)

For \(PQRS\):
\(PQ\): between \((-9,4)\) and \((-6,7)\): \(\sqrt{(-6 + 9)^2 + (7 - 4)^2}=\sqrt{9 + 9}=3\sqrt{2}\)
\(QR\): between \((-6,7)\) and \((-4,4)\): \(\sqrt{(-4 + 6)^2 + (4 - 7)^2}=\sqrt{4 + 9}=\sqrt{13}\)
\(RS\): between \((-4,4)\) and \((-6,2)\): \(\sqrt{(-6 + 4)^2 + (2 - 4)^2}=\sqrt{4 + 4}=2\sqrt{2}\)
\(SP\): between \((-6,2)\) and \((-9,4)\): \(\sqrt{(-9 + 6)^2 + (4 - 2)^2}=\sqrt{9 + 4}=\sqrt{13}\)

Wait, no, that's not matching. Wait, maybe I mixed up the order. Let's check the shape: \(DEFG\) is a quadrilateral with vertices \(D(2,6)\), \(E(4,8)\), \(F(7,6)\), \(G(4,4)\) – this is a rhombus? Wait, \(DE\) and \(FG\) should be equal, \(EF\) and \(GD\) equal. \(DE\): \(2\sqrt{2}\), \(FG\): \(\sqrt{(7 - 4)^2 + (6 - 4)^2}=\sqrt{9 + 4}=\sqrt{13}\)? No, I must have messed up coordinates. Wait, \(G\) is at \((4,4)\), \(F\) at \((7,6)\): \(x\) from 4 to 7 is +3, \(y\) from 4 to 6 is +2, so \(FG=\sqrt{3^2 + 2^2}=\sqrt{13}\). \(D(2,6)\) to \(E(4,8)\): \(x\) +2, \(y\) +2, so \(DE=\sqrt{2^2 + 2^2}=2\sqrt{2}\). \(E(4,8)\) to \(G(4,4)\): \(x\) 0, \(y\) -4, so \(EG = 4\). \(D(2,6)\) to \(G(4,4)\): \(x\) +2, \(y\) -2, so \(DG = 2\sqrt{2}\). \(E(4,8)\) to \(F(7,6)\): \(x\) +3, \(y\) -2, so \(EF=\sqrt{3^2 + (-2)^2}=\sqrt{13}\). \(F(7,6)\) to \(G(4,4)\): \(x\) -3, \(y\) -2, so \(FG=\sqrt{(-3)^2 + (-2)^2}=\sqrt{13}\). So \(DE = DG = 2\sqrt{2}\), \(EF = FG = \sqrt{13}\) – so it's a kite? Wait, no, \(DE\) and \(DG\) are equal, \(EF\) and \(FG\) are equal.

Now \(PQRS\): \(Q(-9,4)\), \(R(-6,7)\), \(S(-4,4)\), \(P(-6,2)\). Let's calculate sides:
\(PQ\): \(Q(-9,4)\) to \(R(-6,7)\): \(x\) +3, \(y\) +3, so \(PQ=\sqrt{3^2 + 3^2}=3\sqrt{2}\)
\(QR\): \(R(-6,7)\) to \(S(-4,4)\): \(x\) +2, \(y\) -3, so \(QR=\sqrt{2^2 + (-3)^2}=\sqrt{13}\)
\(RS\): \(S(-4,4)\) to \(P(-6,2)\): \(x\) -2, \(y\) -2, so \(RS=\sqrt{(-2)^2 + (-2)^2}=2\sqrt{2}\)
\(SP\): \(P(-6,2)\) to \(Q(-9,4)\): \(x\) -3, \(y\) +2, so \(SP=\sqrt{(-3)^2 + 2^2}=\sqrt{13}\)

Wait, now comparing \(DEFG\) and \(PQRS\):
\(DEFG\) sides: \(DE = 2\sqrt{2}\), \(EF = \sqrt{13}\), \(FG = \sqrt{13}\), \(GD = 2\sqrt{2}\) (if order is \(D - E - F - G - D\))
\(PQRS\) sides: \(PQ = 3\sqrt{2}\), \(QR = \sqrt{13}\), \(RS = 2\sqrt{2}\), \(SP = \sqrt{13}\) (order \(Q - R - S - P - Q\))

Wait, no, the side lengths don't match. Wait, maybe I got the order wrong. Let's check the distance between \(D(2,6)\) and \(G(4,4)\): \(\sqrt{(4 - 2)^2 + (4 - 6)^2}=\sqrt{4 + 4}=2\sqrt{2}\). \(E(4,8)\) and \(F(7,6)\): \(\sqrt{(7 - 4)^2 + (6 - 8)^2}=\sqrt{9 + 4}=\sqrt{13}\). \(Q(-9,4)\) and \(S(-4,4)\): distance is \(|-4 - (-9)| = 5\) (horizontal line, \(y\) same). \(R(-6,7)\) and \(P(-6,2)\): distance is \(|7 - 2| = 5\) (vertical line, \(x\) same). Wait, \(DEFG\): \(D(2,6)\), \(G(4,4)\): not horizontal/vertical. \(E(4,8)\), \(F(7,6)\): not horizontal/vertical. Wait, maybe \(DEFG\) has sides: \(DE\): from (2,6) to (4,8): length \(2\sqrt{2}\). \(EF\): (4,8) to (7,6): \(\sqrt{13}\). \(FG\): (7,6) to (4,4): \(\sqrt{13}\). \(GD\): (4,4) to (2,6): \(2\sqrt{2}\). \(PQRS\): \(Q(-9,4)\) to \(R(-6,7)\): \(3\sqrt{2}\). \(R(-6,7)\) to \(S(-4,4)\): \(\sqrt{13}\). \(S(-4,4)\) to \(P(-6,2)\): \(2\sqrt{2}\). \(P(-6,2)\) to \(Q(-9,4)\): \(\sqrt{13}\). Wait, the side lengths of \(DEFG\) are \(2\sqrt{2}, \sqrt{13}, \sqrt{13}, 2\sqrt{2}\). \(PQRS\) has sides \(3\sqrt{2}, \sqrt{13}, 2\sqrt{2}, \sqrt{13}\). Wait, no, \(3\sqrt{2}\) is not equal to \(2\sqrt{2}\). Wait, but maybe I made a mistake in coordinates. Let's re-express coordinates:

\(DEFG\) vertices:
\(D\): (2,6)
\(E\): (4,8)
\(F\): (7,6)
\(G\): (4,4)

\(PQRS\) vertices:
\(Q\): (-9,4)
\(R\): (-6,7)
\(S\): (-4,4)
\(P\): (-6,2)

Now, let's calculate the length of \(DQ\)? No, better to check if one can be translated/rotated/reflected to match the other.

Wait, \(DEFG\): The distance between \(D(2,6)\) and \(F(7,6)\) is \(7 - 2 = 5\) (horizontal line, \(y=6\)). The distance between \(E(4,8)\) and \(G(4,4)\) is \(8 - 4 = 4\) (vertical line, \(x=4\)).

\(PQRS\): The distance between \(Q(-9,4)\) and \(S(-4,4)\) is \(-4 - (-9) = 5\) (horizontal line, \(y=4\)). The distance between \(R(-6,7)\) and \(P(-6,2)\) is \(7 - 2 = 5\) (vertical line, \(x=-6\)). Wait, earlier mistake! \(R(-6,7)\) to \(P(-6,2)\): \(7 - 2 = 5\), not 4. Oh! I see, \(E(4,8)\) to \(G(4,4)\): \(8 - 4 = 4\), but \(R(-6,7)\) to \(P(-6,2)\): \(7 - 2 = 5\). So the vertical diagonals (or the vertical sides) are different. Wait, no, \(DEFG\): \(D(2,6)\), \(F(7,6)\): length 5 (horizontal). \(E(4,8)\), \(G(4,4)\): length 4 (vertical). \(PQRS\): \(Q(-9,4)\), \(S(-4,4)\): length 5 (horizontal). \(R(-6,7)\), \(P(-6,2)\): length 5 (vertical). So the vertical distance in \(DEFG\) is 4, in \(PQRS\) is 5. Therefore, the side lengths related to these will differ.

Wait, let's recalculate \(DE\): \(D(2,6)\) to \(E(4,8)\): \(\sqrt{(4-2)^2 + (8-6)^2} = \sqrt{4 + 4} = 2\sqrt{2}\). \(PQ\): \(Q(-9,4)\) to \(R(-6,7)\): \(\sqrt{(-6+9)^2 + (7-4)^2} = \sqrt{9 + 9} = 3\sqrt{2}\). These are different. So the side lengths are not equal, so the quadrilaterals are not congruent.

Step3: Compare side lengths

Since the side lengths of \(DEFG\) (e.g., \(DE = 2\sqrt{2}\)) do not match the corresponding side lengths of \(PQRS\) (e.g., \(PQ = 3\sqrt{2}\)), and congruent figures must have all corresponding sides equal, \(DEFG\) is not congruent to \(PQRS\).

Answer:

No, \(DEFG\) is not congruent to \(PQRS\). The side lengths (e.g., \(DE = 2\sqrt{2}\) vs. \(PQ = 3\sqrt{2}\)) and vertical/horizontal distances (e.g., \(EG = 4\) vs. \(RP = 5\)) differ, so corresponding sides are not equal, violating the congruence condition.