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
