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

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 \(RS=\sqrt{(0 - 0)^2+(a - 0)^2}=a\).
• For statement 4: If two segments have the same length (from statements 2 and 3 where \(RS=a\) 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 rectangles and squares.

Answer:

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