Question

name: date: period: 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 using distance formula

For \(AB\) with \(A(0,3)\) and \(B(0,0)\), \(AB=\sqrt{(0 - 0)^2+(3 - 0)^2}=3\).
For \(BC\) with \(B(0,0)\) and \(C(-4,0)\), \(BC=\sqrt{(0+4)^2+(0 - 0)^2}=4\).
For \(CA\) with \(C(-4,0)\) and \(A(0,3)\), \(CA=\sqrt{(0 + 4)^2+(3 - 0)^2}=5\).
Since reflection preserves side - lengths, \(DE = AB = 3\), \(EF=BC = 4\), \(FD = CA=5\).

Step2: Identify congruent sides

By reflection, \(\overline{AB}\cong\overline{DE}\), \(\overline{BC}\cong\overline{EF}\), \(\overline{CA}\cong\overline{FD}\).

Step3: Identify congruent angles

By reflection, \(\angle ABC\cong\angle DEF\), \(\angle BCA\cong\angle EFD\), \(\angle CAB\cong\angle FDE\).

Step4: Determine congruent triangles

Since all corresponding sides and angles are congruent, \(\triangle ABC\cong\triangle DEF\).

Step5: Explain using rigid transformation property

A rigid transformation (reflection is a rigid transformation) preserves distances and angles. So if \(\triangle DEF\) is the image of \(\triangle ABC\) under a rigid transformation, \(\triangle ABC\cong\triangle DEF\).

Answer:

• \(AB = 3\), \(BC = 4\), \(CA = 5\), \(DE = 3\), \(EF = 4\), \(FD = 5\)
• \(\overline{AB}\cong\overline{DE}\), \(\overline{BC}\cong\overline{EF}\), \(\overline{CA}\cong\overline{FD}\)
• \(\angle ABC\cong\angle DEF\), \(\angle BCA\cong\angle EFD\), \(\angle CAB\cong\angle FDE\)
• \(\triangle ABC\cong\triangle DEF\)
• A rigid transformation (reflection) preserves the shape and size of the figure (preserves side - lengths and angle measures).