To complete the proof that [tex]\( RSTU \)[/tex] is a square, we need to provide the correct order of reasons for the given statements. Let's go through the solution step by step:
1. RSTU is a rectangle with vertices [tex]\( R(0,0), S(0,a), T(a,a), \)[/tex] and [tex]\( U(a,0) \)[/tex]:
- Reason: Given
2. [tex]\( RS = a \)[/tex] units:
- Reason: Distance formula. We use the distance formula to calculate the distance between points [tex]\( R(0,0) \)[/tex] and [tex]\( S(0,a) \)[/tex]:
[tex]\[
RS = \sqrt{(0-0)^2 + (a-0)^2} = \sqrt{a^2} = a
\][/tex]
3. [tex]\( ST = a \)[/tex] units:
- Reason: Distance formula. We use the distance formula to calculate the distance between points [tex]\( S(0,a) \)[/tex] and [tex]\( T(a,a) \)[/tex]:
[tex]\[
ST = \sqrt{(a-0)^2 + (a-a)^2} = \sqrt{a^2} = a
\][/tex]
4. [tex]\( \overline{RS} \cong \overline{ST} \)[/tex]:
- Reason: Definition of congruence. Since [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] are both equal to [tex]\( a \)[/tex], they are congruent. Congruence of line segments is defined by having equal lengths.
5. RSTU is a square:
- Reason: If two consecutive sides of a rectangle are congruent, then it's a square. We have shown that [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex], which are consecutive sides of the rectangle RSTU, are congruent. This means that RSTU must be a square.
From these steps, the correct order of reasons is:
C. distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square.
