Question

complete each problem as indicated. you must show all of your work to receive full credit.

1. graph \\( \triangle a b c \\) with vertices \\( a ( 0,3 ), b ( 0,0 ) \\) and \\( c ( - 4,0 ) \\) on the graph provided.

a. reflect \\( \triangle a b c \\) across \\( \overline { a b } \\) to construct \\( \triangle d e f \\).
b. list the side lengths for all segments.
\\( a b = \\)
\\( d e = \\)
\\( b c = \\)
\\( e f = \\)
\\( c a = \\)
\\( f d = \\)
mark the sides that are congruent on the graph. list all sides that are congruent.
\\( \overline { a b } \cong \\)
\\( \overline { b c } \cong \\)
\\( \overline { c a } \cong \\)
d. verify angles are congruent using a protractor. mark the angles that are congruent on the graph. list all angles that are congruent.
\\( \angle a b c \cong \\)
\\( \angle b c a \cong \\)
\\( \angle c a b \cong \\)
e. if all sides and angles of two triangles are congruent, then the two triangles are congruent. therefore,
\\( \delta \cong \delta \\)
f. assume we were not able to measure the sides and angles of \\( \triangle a b c \\) and \\( \triangle d e f \\). knowing \\( \triangle d e f \\) is the image of \\( \triangle a b c \\) resulting from a rigid transformation would have been sufficient to determine \\( \triangle a b c \cong \triangle d e f \\), because

Explanation:

Step1: Calculate side lengths

• For \(AB\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(A(0,3)\) and \(B(0,0)\), \(AB=\sqrt{(0 - 0)^2+(0 - 3)^2}=3\)
• For \(BC\): With \(B(0,0)\) and \(C(-4,0)\), \(BC=\sqrt{(-4 - 0)^2+(0 - 0)^2}=4\)
• For \(CA\): With \(C(-4,0)\) and \(A(0,3)\), \(CA=\sqrt{(0+4)^2+(3 - 0)^2}=5\)
• Since \(\triangle DEF\) is a reflection of \(\triangle ABC\) across \(AB\), \(DE = AB = 3\), \(EF=BC = 4\), \(FD=CA = 5\)

Step2: Determine congruent sides

• By reflection (a rigid transformation), \(\overline{AB}\cong\overline{DE}\), \(\overline{BC}\cong\overline{EF}\), \(\overline{CA}\cong\overline{FD}\)

Step3: Determine congruent angles

• \(\angle ABC\cong\angle DEF\) (right angles), \(\angle BCA\cong\angle EFD\), \(\angle CAB\cong\angle FDE\)

Step4: Congruent triangles

• \(\triangle ABC\cong\triangle DEF\)

Step5: Rigid transformation property

• A rigid transformation (reflection in this case) preserves side - lengths and angle - measures. So if \(\triangle DEF\) is the image of \(\triangle ABC\) under a rigid transformation, then \(\triangle ABC\cong\triangle DEF\)

Answer:

b. \(AB = 3\), \(DE = 3\), \(BC = 4\), \(EF = 4\), \(CA = 5\), \(FD = 5\)
c. \(\overline{AB}\cong\overline{DE}\), \(\overline{BC}\cong\overline{EF}\), \(\overline{CA}\cong\overline{FD}\)
d. \(\angle ABC\cong\angle DEF\), \(\angle BCA\cong\angle EFD\), \(\angle CAB\cong\angle FDE\)
e. \(\triangle ABC\cong\triangle DEF\)
f. A rigid transformation (like reflection) preserves the shape and size (side - lengths and angle - measures) of a figure. So the two triangles are congruent.