To prove that RSTU is a square, we need to determine the lengths of its sides and analyze the relationships between consecutive sides. Here's the step-by-step reasoning for each statement:
Given that RSTU is a rectangle with vertices [tex]\( R(0,0) \)[/tex], [tex]\( S(0,a) \)[/tex], [tex]\( T(a,a) \)[/tex], and [tex]\( U(a,0) \)[/tex]:
1. Statement: [tex]\( RSTU \)[/tex] is a rectangle with vertices [tex]\( R(0,0) \)[/tex], [tex]\( S(0,a) \)[/tex], [tex]\( T(a,a) \)[/tex], and [tex]\( U(a,0) \)[/tex].
- Reason: Given.
2. Statement: [tex]\( \overline{RS} = a \)[/tex] units.
- Reason: Distance formula. The distance between [tex]\( R(0,0) \)[/tex] and [tex]\( S(0,a) \)[/tex] is [tex]\(\sqrt{ (0-0)^2 + (a-0)^2 } = \sqrt{ a^2 } = a \)[/tex].
3. Statement: [tex]\( \overline{ST} = a \)[/tex] units.
- Reason: Distance formula. The distance between [tex]\( S(0,a) \)[/tex] and [tex]\( T(a,a) \)[/tex] is [tex]\(\sqrt{ (a-0)^2 + (a-a)^2 } = \sqrt{ a^2 } = a \)[/tex].
4. Statement: [tex]\( \overline{RS} \cong \overline{ST} \)[/tex].
- Reason: Definition of congruence. Since the lengths of [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{ST} \)[/tex] are equal ([tex]\(a\)[/tex] units), the segments [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{ST} \)[/tex] are congruent.
5. Statement: [tex]\( RSTU \)[/tex] is a square.
- Reason: If two consecutive sides of a rectangle are congruent, then it's a square.
Hence, the correct order of reasons that complete the proof is:
B. distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square.
