Question

Triangles ABE, ADE, and CBE are shown on the coordinate grid, and all the vertices have coordinates that are integers00С2E3402A-2Which statement is true?No two triangles are congruent.Only Triangles ABE and CBE are congruent.only triangles ABE and ADE are congruenttriangles ABE, ADE, and CBE are all congruent

Answer 1

AThe coordinates of the triangles are given below

`\begin{gathered} A=(-4,-1) \\ B=(-1,3) \\ C=(4,3) \\ D=(-1,1) \\ E=(0,1) \end{gathered}`

Considering the two triangles ABE and CBE

`\begin{gathered} AB=\sqrt[]{(-1-(-4)^2+(3-(-1)^2} \\ AB=\sqrt[]{3^2+4^2} \\ AB=\sqrt[]{9+16} \\ AB=\sqrt[]{25} \\ AB=5 \end{gathered}`
`\begin{gathered} AE=\sqrt[]{(0-(-4)^2+(1-(-1)^2} \\ AE=\sqrt[]{4^2+2^2} \\ AE=\sqrt[]{16+4} \\ AE=\sqrt[]{20} \end{gathered}`
`\begin{gathered} BE=\sqrt[]{(0-(-1)^2+(1-3)^2} \\ BE=\sqrt[]{1^2+(-2)^2} \\ BE=\sqrt[]{1+4} \\ BE=\sqrt[]{5} \end{gathered}`
`\begin{gathered} CB=\sqrt[]{(4-(-1)^2+(3-3)^2} \\ CB=\sqrt[]{(4+1)^2}+0 \\ CB=5 \end{gathered}`
`\begin{gathered} CE=\sqrt[]{(4-0)^2+}((3-1)^2 \\ CE=\sqrt[]{4^2+2^2} \\ CE=\sqrt[]{16+4} \\ CE=\sqrt[]{20} \end{gathered}`
`\begin{gathered} BE=\sqrt[]{(-1-0)^2+(3-1)^2} \\ BE=\sqrt[]{(-1)^2+2^2} \\ BE=\sqrt[]{1+4} \\ BE=\sqrt[]{5} \end{gathered}`

Consider triangle ADE

`\begin{gathered} AE=\sqrt[]{(0-(-4)^2+(1-(-1)^2} \\ AE=\sqrt[]{4^2+2^2} \\ AE=\sqrt[]{16+4} \\ AE=\sqrt[]{20} \end{gathered}`
`\begin{gathered} AD=\sqrt[]{(-4-1)^2+(-1-(-1)^2} \\ AD=\sqrt[]{(-5)^2} \\ AD=\sqrt[]{25} \\ AD=5 \end{gathered}`
`\begin{gathered} DE=\sqrt[]{(1-0)^2+(-1-1)^2} \\ DE=\sqrt[]{1^2+(-2)^2} \\ DE=\sqrt[]{1+4} \\ DE=\sqrt[]{5} \end{gathered}`

From the calculation above,

we can conclude that ABE, CBE and ADE are all CONGRUENT