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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].

Prove: RSTU is a square.

[tex]\[
\begin{array}{|l|l|}
\hline \text{Statements} & \text{Reasons} \\
\hline 1. \ RSTU \ \text{is a rectangle with vertices} \ R(0,0), S(0, a), T(a, a), \ \text{and} \ U(a, 0) & 1. \ \text{Given} \\
\hline 2. \ RS = a \ \text{units} & 2. \ ? \\
\hline 3. \ ST = a \ \text{units} & 3. \ \text{Distance formula} \\
\hline 4. \ \overline{RS} \cong \overline{ST} & 4. \ ? \\
\hline 5. \ RSTU \ \text{is a square} & 5. \ ? \\
\hline
\end{array}
\][/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. Definition of congruence; distance formula; 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. Distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square



Answer :

GinnyAnswer
Let's provide a detailed, step-by-step solution to determine the correct order of reasons that complete the proof.

1. Given: RSTU is a rectangle with vertices [tex]\( R(0,0), S(0, a), T(a, a) \)[/tex], and [tex]\( U(a, 0) \)[/tex]:
- This is stated as a given fact.

2. Prove: RS = [tex]\( a \)[/tex] units:
- [tex]\( R(0,0) \)[/tex] and [tex]\( S(0,a) \)[/tex] have the same x-coordinate and different y-coordinates.
- To find the distance [tex]\( RS \)[/tex], subtract the y-coordinates: [tex]\( |a - 0| = a \)[/tex].
- Therefore, [tex]\( RS = a \)[/tex].

- Reason: Distance formula for vertical or horizontal segments reduces to the absolute difference in coordinates of the two points. Alternatively, one could just infer the length from the coordinates directly since it's simply the difference in y-coordinates for vertical segments.

3. Prove: ST = [tex]\( a \)[/tex] units:
- [tex]\( S(0,a) \)[/tex] and [tex]\( T(a,a) \)[/tex] have the same y-coordinate and different x-coordinates.
- To find the distance [tex]\( ST \)[/tex], subtract the x-coordinates: [tex]\( |a - 0| = a \)[/tex].
- Therefore, [tex]\( ST = a \)[/tex].

- Reason: This uses the standard distance formula applied to horizontal segments: the difference in x-coordinates.

4. Prove: [tex]\( \overline{RS} \cong \overline{ST} \)[/tex]:
- Both [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] are equal to [tex]\( a \)[/tex].
- Therefore, [tex]\( RS \)[/tex] is congruent to [tex]\( ST \)[/tex].

- Reason: Definition of congruence of segments, which states that segments are congruent if they have equal length.

5. Conclude: RSTU is a square:
- In a rectangle, if two consecutive sides are congruent, then all four sides must be equal.
- Hence, since [tex]\( RS = ST \)[/tex] and both are sides of the rectangle, RSTU must be a square.

- Reason: If two consecutive sides of a rectangle are congruent, it must be a square.

Thus, the correct order of reasons that complete the proof is:
1. Given
2. Distance formula (to find lengths of RS and ST)
3. Definition of congruence (to establish [tex]\( \overline{RS} \cong \overline{ST} \)[/tex])
4. If two consecutive sides of a rectangle are congruent, then it's a square (to conclude it's a square)

The correct choice is:

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

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