To prove that RSTU is a square, given the vertices [tex]\( R(0,0) \)[/tex], [tex]\( S(0,a) \)[/tex], [tex]\( T(a,a) \)[/tex], and [tex]\( U(a,0) \)[/tex], we need to verify that all sides of the rectangle are congruent (equal in length).
1. Calculate the lengths of the sides:
- RS (distance between [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]
- ST (distance between [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]
- TU (distance between [tex]\(T(a,a)\)[/tex] and [tex]\(U(a,0)\)[/tex]):
[tex]\[
TU = \sqrt{(a-a)^2 + (0-a)^2} = \sqrt{a^2} = a
\][/tex]
- UR (distance between [tex]\(U(a,0)\)[/tex] and [tex]\(R(0,0)\)[/tex]):
[tex]\[
UR = \sqrt{(0-a)^2 + (0-0)^2} = \sqrt{a^2} = a
\][/tex]
2. Use the definition of congruence:
According to the definition of congruence, if two segments have the same length, they are congruent.
3. Check if two consecutive sides of the rectangle are congruent:
Since [tex]\( RS = ST = TU = UR = a \)[/tex], all four sides of the rectangle are congruent.
4. Determine the shape:
By the property that if two consecutive sides of a rectangle are congruent, then that rectangle is a square.
Therefore, the correct order to complete the proof is:
1. Calculate lengths using the distance formula,
2. Apply the definition of congruence,
3. Conclude that if two consecutive sides of a rectangle are congruent, then it's a square.
Hence, the correct answer is:
D. distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square
