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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.

\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]\( R(0,0), S(0, a), T(a, a), \)[/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



Answer :

GinnyAnswer
To prove that RSTU is a square given the 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], we need to establish a sequence of logical statements and corresponding reasons.

Let's break down the steps and apply the correct reasoning in each step.

### Statements and Reasons:

1. Given: 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].

2. Determine RS:
- Statement: [tex]\( RS = a \)[/tex] units.
- Reason: This can be directly observed from the coordinates given since [tex]\( S \)[/tex] has coordinates [tex]\( (0, a) \)[/tex] and [tex]\( R \)[/tex] has coordinates [tex]\( (0, 0) \)[/tex]. [tex]\( RS \)[/tex] is simply the vertical distance between these points, which is [tex]\( a \)[/tex].

3. Determine ST:
- Statement: [tex]\( ST = a \)[/tex] units.
- Reason: We can find the length of [tex]\( ST \)[/tex] using the distance formula: [tex]\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]. Here, [tex]\( S(0, a) \)[/tex] and [tex]\( T(a, a) \)[/tex], so [tex]\( ST = \sqrt{(a - 0)^2 + (a - a)^2} = \sqrt{a^2 + 0} = a \)[/tex].

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

5. Conclusion about RSTU:
- Statement: RSTU is a square.
- Reason: If two consecutive sides of a rectangle are congruent, then the rectangle is a square.

### Correct Sequence of Reasons:

Reviewing the steps logically, we have:
1. Given.
2. Direct observation (proportionality of length in coordinate geometry).
3. Distance formula.
4. Definition of congruence.
5. If two consecutive sides of a rectangle are congruent, then it's a square.

The correct order of reasons is:

1. Given.
2. Direct observation (length in coordinate geometry).
3. Distance formula.
4. Definition of congruence.
5. If two consecutive sides of a rectangle are congruent, then it’s a square.

The corresponding option 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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