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Given: [tex]\( RSTU \)[/tex] is a rectangle with vertices [tex]\( R(0,0), S(0, a), T(a, a) \)[/tex], and [tex]\( U(a, 0) \)[/tex], where [tex]\( a \neq 0 \)[/tex].

Prove: [tex]\( RSTU \)[/tex] is a square.

[tex]\[
\begin{tabular}{|l|l|}
\hline
\multicolumn{1}{|c|}{Statements} & Reasons \\
\hline
1. \( RSTU \) is a rectangle with vertices \( R (0,0), S (0, a ), T ( a , a ) \), and \( U ( a , 0) \) & 1. Given \\
\hline
2. \( RS = a \) units & 2. Definition of length of a vertical segment \\
\hline
3. \( ST = a \) units & 3. Distance formula \\
\hline
4. \( \overline{RS} \cong \overline{ST} \) & 4. Definition of congruence \\
\hline
5. \( RSTU \) is a square. & 5. If two consecutive sides of a rectangle are congruent, then it is a square. \\
\hline
\end{tabular}
\][/tex]

What is the correct order of reasons that complete the proof?

A. If two consecutive sides of a rectangle are congruent, then it's a square; distance formula; definition of congruence
B. Distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square
C. Distance formula; if two consecutive sides of a rectangle are congruent, then it's a square; definition of congruence
D. Definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it's a square

Sagot :

GinnyAnsweridn
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.
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