Question

select the correct answer.
given: rstu is a rectangle with vertices r(0,0), s(0, a), t(a, a), and u(a, 0), where a ≠ 0.
prove: rstu is a square.

statements	reasons
2. rs = a units	2.?
3. st = a units	3. distance formula
4. \\(\overline{rs} \cong \overline{st}\\)	4.?
5. rstu is a square.	5.?

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. if two consecutive sides of a rectangle are congruent, then it’s a square; distance formula; definition of congruence
c. definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it’s a square
d. distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it’s a square

Explanation:

1. For step 2 (finding \( RS \)), we use the distance formula. The distance between \( R(0,0) \) and \( S(0,a) \) is calculated using the distance formula \( d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(0 - 0)^2+(a - 0)^2}=a \), so the reason for step 2 is the distance formula.
2. For step 4 (\( \overline{RS}\cong\overline{ST} \)), since \( RS = a \) and \( ST=a \), by the definition of congruence (if two segments have the same length, they are congruent), we can say \( \overline{RS}\cong\overline{ST} \).
3. For step 5 (proving RSTU is a square), the theorem "if two consecutive sides of a rectangle are congruent, then it's a square" is used. Since RSTU is a rectangle and two consecutive sides \( RS \) and \( ST \) are congruent, RSTU is a square. So the order of reasons is distance formula (step 2), definition of congruence (step 4), and the rectangle - to - square theorem (step 5), which matches option D.

Answer:

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