Question

identifying a congruent triangle
which are the coordinates of the vertices of a triangle congruent to (\triangle pqr)?
( (0,5),(2,10),(0,10) )
( (9,-3),(9,0),(4,0) )
( (4,3),(-4,8),(-4,3) )
( (2.5,-4),(0,0),(0,-4) )

Explanation:

Step1: Calculate the side lengths of \(\triangle PQR\)

• For \(PQ\): Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(P(0,0)\) and \(Q(0, - 8)\), \(d_{PQ}=\sqrt{(0 - 0)^2+(-8 - 0)^2}=8\)
• For \(QR\): With \(Q(0,-8)\) and \(R(5,-8)\), \(d_{QR}=\sqrt{(5 - 0)^2+(-8+8)^2}=5\)
• For \(PR\): With \(P(0,0)\) and \(R(5,-8)\), \(d_{PR}=\sqrt{(5 - 0)^2+(-8 - 0)^2}=\sqrt{25 + 64}=\sqrt{89}\)

Step2: Calculate the side lengths of each option

• Option 1: Let \(A(0,5)\), \(B(2,10)\), \(C(0,10)\)
• \(d_{AB}=\sqrt{(2 - 0)^2+(10 - 5)^2}=\sqrt{4 + 25}=\sqrt{29}\)
• \(d_{BC}=\sqrt{(0 - 2)^2+(10 - 10)^2}=2\)
• \(d_{AC}=\sqrt{(0 - 0)^2+(10 - 5)^2}=5\)
• Option 2: Let \(D(9,-3)\), \(E(9,0)\), \(F(4,0)\)
• \(d_{DE}=\sqrt{(9 - 9)^2+(0 + 3)^2}=3\)
• \(d_{EF}=\sqrt{(4 - 9)^2+(0 - 0)^2}=5\)
• \(d_{DF}=\sqrt{(4 - 9)^2+(0 + 3)^2}=\sqrt{25+9}=\sqrt{34}\)
• Option 3: Let \(G(4,3)\), \(H(-4,8)\), \(I(-4,3)\)
• \(d_{GH}=\sqrt{(-4 - 4)^2+(8 - 3)^2}=\sqrt{64 + 25}=\sqrt{89}\)
• \(d_{HI}=\sqrt{(-4+4)^2+(3 - 8)^2}=5\)
• \(d_{GI}=\sqrt{(-4 - 4)^2+(3 - 3)^2}=8\)
• Option 4: Let \(J(2.5,-4)\), \(K(0,0)\), \(L(0,-4)\)
• \(d_{JK}=\sqrt{(0 - 2.5)^2+(0 + 4)^2}=\sqrt{6.25 + 16}=\sqrt{22.25}\)
• \(d_{KL}=\sqrt{(0 - 0)^2+(-4 - 0)^2}=4\)
• \(d_{JL}=\sqrt{(0 - 2.5)^2+(-4 + 4)^2}=2.5\)

Answer:

\((4,3),(-4,8),(-4,3)\)