Question

the graph shows triangles jkl and qrs. is jkl congruent to qrs? justify your answer.

Explanation:

Step1: Find coordinates of vertices

• \( J(-4, -3) \), \( K(-2, -7) \), \( L(-9, -6) \)
• \( Q(3, -4) \), \( R(7, -2) \), \( S(6, -9) \)

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

• \( JK \): \( \sqrt{(-2 + 4)^2 + (-7 + 3)^2} = \sqrt{4 + 16} = \sqrt{20} \)
• \( KL \): \( \sqrt{(-9 + 2)^2 + (-6 + 7)^2} = \sqrt{49 + 1} = \sqrt{50} \)
• \( LJ \): \( \sqrt{(-4 + 9)^2 + (-3 + 6)^2} = \sqrt{25 + 9} = \sqrt{34} \)
• \( QR \): \( \sqrt{(7 - 3)^2 + (-2 + 4)^2} = \sqrt{16 + 4} = \sqrt{20} \)
• \( RS \): \( \sqrt{(6 - 7)^2 + (-9 + 2)^2} = \sqrt{1 + 49} = \sqrt{50} \)
• \( SQ \): \( \sqrt{(3 - 6)^2 + (-4 + 9)^2} = \sqrt{9 + 25} = \sqrt{34} \)

Step3: Compare side lengths

• \( JK = QR = \sqrt{20} \), \( KL = RS = \sqrt{50} \), \( LJ = SQ = \sqrt{34} \)
• By SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Answer:

Yes, \( \triangle JKL \cong \triangle QRS \) by SSS congruence (all corresponding sides are equal).