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Given: [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], where [tex]\(a \neq 0\)[/tex].

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

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

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

A. Distance formula; if two consecutive sides of a rectangle are congruent, then it's a square; definition of congruence

B. Distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square

C. Definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it's a square

D. If two consecutive sides of a rectangle are congruent, then it's a square; distance formula; definition of congruence



Answer :

GinnyAnswer
To solve this problem, we need to prove that RSTU is a square and determine the correct order of reasons to complete the proof. Here is the detailed step-by-step solution:

Given:
- RSTU 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].

To Prove:
- RSTU is a square.

Proof:
1. Statement 1: [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].
- Reason 1: given

2. Statement 2: [tex]\( RS = a \)[/tex] units
- Reason 2: distance formula

To find the length of [tex]\( RS \)[/tex]:
[tex]\[ RS = \sqrt{(0-0)^2 + (a-0)^2} = \sqrt{0 + a^2} = \sqrt{a^2} = a \text{ units} \][/tex]

3. Statement 3: [tex]\( ST = a \)[/tex] units
- Reason 3: distance formula

To find the length of [tex]\( ST \)[/tex]:
[tex]\[ ST = \sqrt{(a-0)^2 + (a-a)^2} = \sqrt{a^2 + 0} = \sqrt{a^2} = a \text{ units} \][/tex]

4. Statement 4: [tex]\( \overline{RS} \cong \overline{ST} \)[/tex]
- Reason 4: definition of congruence (two segments are congruent if they have the same length)

5. Statement 5: [tex]\( RSTU \)[/tex] is a square.
- Reason 5: if two consecutive sides of a rectangle are congruent, then it's a square

Since [tex]\( RSTU \)[/tex] is given as a rectangle and we have proven that two consecutive sides [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] are congruent ([tex]\( RS = ST = a \)[/tex]), it follows that [tex]\( RSTU \)[/tex] is a square.

Conclusion:
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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