Question

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.

what is the correct order of reasons that complete the proof?
a. distance formula, definition of congruence, if two consecutive sides of a rectangle are congruent, then its a square
b. if two consecutive sides of a rectangle are congruent, then its a square; distance formula, definition of congruence
c. definition of congruence, distance formula; if two consecutive sides of a rectangle are congruent, then its a square
d. distance formula; if two consecutive sides of a rectangle are congruent, then its a square; definition of congruence

Explanation:

• For statement 2: To find the length of \(RS\) with \(R(0,0)\) and \(S(0,a)\), we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 0,y_1=0,x_2 = 0,y_2=a\), so \(RS=\sqrt{(0 - 0)^2+(a - 0)^2}=a\) (distance formula).
• For statement 4: Congruent segments have equal lengths. Since \(RS=a\) and \(ST=a\), by the definition of congruence (\(\overline{AB}\cong\overline{CD}\) if \(AB = CD\)), \(\overline{RS}\cong\overline{ST}\).
• For statement 5: A rectangle is a square if two - consecutive sides are congruent. This is a well - known property of squares (if two consecutive sides of a rectangle are congruent, then it's a square).

Answer:

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