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Which statement correctly names the congruent triangles and justifies the reason for congruence? a coordinate plane with triangles ABC and FED at A 0 comma 0, B 4 comma 0, C 1 comma 3, D 5 comma 2, E 8 comma 5, and F 8 comma 1, segment AC measures 3 and 2 tenths units, angle A measures 72 degrees, segment DF measures 3 and 2 tenths units, and angle F measures 72 degrees Î"ABC ≅ Î"FDE by HL Î"ABC ≅ Î"FED by SAS Î"ABC ≅ Î"DEF by HL Î"ABC ≅ Î"DEF by SAS.

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Use the distance formula to justify the congruence. The correct option is B. [tex]\Delta ABC\cong \Delta FED\\[/tex].

Given,

the coordinates of [tex]\Delta ABC[/tex] and [tex]\Delta FED[/tex] are [tex]A(0,0)\ B(4,0)\ C(1,3)\ D(5,2)\ E(8,5)\ F(8,1).[/tex]

The length of segment AC is [tex]\frac{3}{2}[/tex] and the length of segment DF is also [tex]\frac{3}{2}[/tex] .

[tex]\angle A = 72^{\circ}[/tex] and [tex]\angle F =72^{\circ}[/tex].

We have to find the correct statement regarding the triangles to be congruent.

Distance formula

We know that, The distance formula to calculate the distance between two points [tex]( x_{1} , y_{1})[/tex] , and [tex]( x_{2}, y_{2})[/tex] is given as,

[tex]D=\sqrt{(x_{2}-x_{1} ) ^{2}+(y_{2} -y_{1} )^2 }[/tex]

So, The distance between [tex]F(8,1)[/tex] and [tex]E(8,5)[/tex] Will be,

[tex]FE=\sqrt{(8-8)^2+(5-1)^2} \\FE=\sqrt{4^2} \\FE=4\\[/tex]

Similarly, the distance between [tex]A(0,0)[/tex] and [tex]B(4,0)[/tex] will be,

[tex]AB=\sqrt{(4-0)^2+(0-0)^2} \\AB=\sqrt{4^2} \\AB=4[/tex]

So, [tex]AB=FE[/tex].......(1)

Now, in [tex]\Delta ABC[/tex] and [tex]\Delta FED[/tex],

[tex]AB=FE[/tex] ( from equation 1 ).

[tex]\angle A=\angle F[/tex] ( Given )

[tex]AC=DF[/tex] ( Given )

So, By Side angle Side congruence rule,

[tex]\Delta ABC\cong \Delta FED[/tex].

Hence the correct option is B. [tex]\Delta ABC\cong \Delta FED\\[/tex].

For more details on SAS congruence rule follow the link:

https://brainly.com/question/11804042

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