Question

is defg congruent to pqrs? justify your answer.

Explanation:

Step1: Find Coordinates of PQRS

- \( Q(-9, 4) \), \( R(-6, 7) \), \( S(-3, 4) \), \( P(-6, 2) \)
- Calculate side lengths (e.g., \( QR \)):
\( QR = \sqrt{(-6 - (-9))^2 + (7 - 4)^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \)
\( RS = \sqrt{(-3 - (-6))^2 + (4 - 7)^2} = \sqrt{3^2 + (-3)^2} = 3\sqrt{2} \)
\( SP = \sqrt{(-6 - (-3))^2 + (2 - 4)^2} = \sqrt{(-3)^2 + (-2)^2} \)? Wait, no—wait, PQRS is a rhombus? Wait, recheck coordinates. Wait, \( Q(-9, 4) \), \( R(-6, 7) \), \( S(-3, 4) \), \( P(-6, 2) \). So \( QR \): from \( (-9,4) \) to \( (-6,7) \): \( \Delta x = 3 \), \( \Delta y = 3 \), length \( \sqrt{3^2 + 3^2} = 3\sqrt{2} \). \( RS \): \( (-6,7) \) to \( (-3,4) \): \( \Delta x = 3 \), \( \Delta y = -3 \), length \( 3\sqrt{2} \). \( SP \): \( (-3,4) \) to \( (-6,2) \): \( \Delta x = -3 \), \( \Delta y = -2 \)? Wait, no, \( P \) is \( (-6, 2) \), so \( S(-3,4) \) to \( P(-6,2) \): \( \Delta x = -3 \), \( \Delta y = -2 \)? Wait, that can't be. Wait, maybe I misread the grid. Let's count grid squares. From \( Q(-9,4) \) to \( R(-6,7) \): 3 right, 3 up (so length \( \sqrt{3^2 + 3^2} = 3\sqrt{2} \)). From \( R(-6,7) \) to \( S(-3,4) \): 3 right, 3 down (length \( 3\sqrt{2} \)). From \( S(-3,4) \) to \( P(-6,2) \): 3 left, 2 down? No, wait \( P \) is at \( x=-6 \), \( y=2 \). So \( S(-3,4) \) to \( P(-6,2) \): \( x \) difference \( -3 \), \( y \) difference \( -2 \). Wait, that's not 3-3. Wait, maybe the figure is a rhombus, so all sides equal. Wait, maybe my coordinate reading is wrong. Let's check the left figure: \( Q \) is at \( x=-9 \) (since -10, -9, ...), \( y=4 \). \( R \) at \( x=-6 \), \( y=7 \). \( S \) at \( x=-3 \), \( y=4 \). \( P \) at \( x=-6 \), \( y=2 \). So the vertical distance from \( R \) to \( P \) is \( 7 - 2 = 5 \)? No, \( R(-6,7) \) to \( P(-6,2) \): vertical line, length 5. Horizontal distance from \( Q \) to \( S \): \( -3 - (-9) = 6 \), so horizontal diagonal is 6, vertical diagonal is 5 (from \( R(-6,7) \) to \( P(-6,2) \): 5 units).

Step2: Find Coordinates of DEFG

- \( D(2,6) \), \( E(4,8) \), \( F(7,6) \), \( G(4,4) \)
- Calculate side lengths (e.g., \( DE \)):
\( DE = \sqrt{(4 - 2)^2 + (8 - 6)^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \)
\( EF = \sqrt{(7 - 4)^2 + (6 - 8)^2} = \sqrt{3^2 + (-2)^2} \)? Wait, no—wait, \( D(2,6) \), \( E(4,8) \): \( \Delta x = 2 \), \( \Delta y = 2 \), length \( 2\sqrt{2} \). \( E(4,8) \) to \( F(7,6) \): \( \Delta x = 3 \), \( \Delta y = -2 \)? No, \( F(7,6) \), so \( E(4,8) \) to \( F(7,6) \): \( \Delta x = 3 \), \( \Delta y = -2 \)? Wait, no, grid count: \( D(2,6) \) to \( E(4,8) \): 2 right, 2 up (length \( 2\sqrt{2} \)). \( E(4,8) \) to \( F(7,6) \): 3 right, 2 down? No, \( F \) is at \( x=7 \), \( y=6 \). So \( E(4,8) \) to \( F(7,6) \): \( \Delta x = 3 \), \( \Delta y = -2 \). Wait, but \( D(2,6) \) to \( G(4,4) \): \( \Delta x = 2 \), \( \Delta y = -2 \), length \( 2\sqrt{2} \). \( G(4,4) \) to \( F(7,6) \): \( \Delta x = 3 \), \( \Delta y = 2 \), length \( \sqrt{3^2 + 2^2} = \sqrt{13} \)? No, this is confusing. Wait, better to check diagonals. For PQRS: horizontal diagonal (Q to S): from \( x=-9 \) to \( x=-3 \), so length 6 (since \( -3 - (-9) = 6 \)). Vertical diagonal (R to P): from \( y=7 \) to \( y=2 \), length 5 (since \( 7 - 2 = 5 \)). For DEFG: horizontal diagonal (D to F): from \( x=2 \) to \( x=7 \), length \( 7 - 2 = 5 \)? Wait, \( D(2,6) \), \( F(7,6) \): \( x \) from 2 to 7 is 5 units (horizontal distance 5). Vertical diagonal (E to G): from \( y=8 \) to \( y=4 \), length \( 8 - 4 = 4 \) (vertical distance 4). Wait, no—wait \( E(4,8) \), \( G(4,4) \): vertical distance 4.

Step3: Compare Diagonals (Rhombus Congruence)

- For a rhombus, congruence requires both diagonals to be equal (since all sides are equal, but diagonals determine size).
- PQRS: horizontal diagonal length \( 6 \) (from \( x=-9 \) to \( x=-3 \): \( -3 - (-9) = 6 \)), vertical diagonal length \( 5 \) (from \( y=7 \) to \( y=2 \): \( 7 - 2 = 5 \)).
- DEFG: horizontal diagonal length \( 5 \) (from \( x=2 \) to \( x=7 \): \( 7 - 2 = 5 \)), vertical diagonal length \( 4 \) (from \( y=8 \) to \( y=4 \): \( 8 - 4 = 4 \)).
- Since the diagonals of PQRS (6 and 5) do not match DEFG (5 and 4), the rhombuses are not congruent. Alternatively, count side lengths: PQRS has side length \( \sqrt{3^2 + 3^2} = 3\sqrt{2} \) (from Q to R: 3 right, 3 up), DEFG has side length \( \sqrt{2^2 + 2^2} = 2\sqrt{2} \) (from D to E: 2 right, 2 up). Different side lengths, so not congruent.

Answer:

No, DEFG is not congruent to PQRS. The side lengths (or diagonals) of the two rhombuses are different: PQRS has side length \( 3\sqrt{2} \) (from 3-unit horizontal/vertical shifts) and DEFG has side length \( 2\sqrt{2} \) (from 2-unit shifts), so their corresponding sides are not equal, violating the congruence condition.