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 $a \neq 0$.
Prove: RSTU is a square.
| Statements | Reasons |
| ----------- | ----------- |
| 1. RSTU is a rectangle with vertices R(0,0), S(0, a), T(a, a), and U(a, 0). | 1. given |
| 2. $R S= a$ units | 2. ? |
| 3. $S T= a$ 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. If two consecutive sides of a rectangle are congruent, then it's a square; distance formula; definition of congruence
B. distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square
C. distance formula; if two consecutive sides of a rectangle are congruent, then ir's a square; definition of congruence
D. definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it's a square
Answer (2)
The rectangle RSTU can be proven to be a square by demonstrating that its sides are equal using the distance formula, establishing congruence between the sides, and applying the property that states if two consecutive sides are congruent, the rectangle is a square. The correct order of reasons that complete the proof corresponds to option B. Therefore, option B is the answer.
;
Use the distance formula to show that RS = a .
Use the definition of congruence to show that RS ≅ ST .
Use the property that if two consecutive sides of a rectangle are congruent, then it is a square.
Conclude that the correct order of reasons is distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square, which corresponds to option B. B
Explanation
Analyze the Given Information Let's analyze the given statements and reasons to determine the correct order of the missing reasons in the proof.
Statement 1: RSTU is a rectangle with vertices R(0,0), S(0, a), T(a, a), and U(a, 0). Reason 1: given This is the starting point of our proof, which is given to us.
Determine Reason 2 Statement 2: RS = a units. Reason 2: ? To find the distance RS, we use the distance formula between points R(0,0) and S(0,a): RS = ( 0 − 0 ) 2 + ( a − 0 ) 2 = 0 + a 2 = a 2 = ∣ a ∣ Since we are given that a = 0 , we can say RS = a . Therefore, the reason for statement 2 is the distance formula.
Acknowledge Reason 3 Statement 3: ST = a units. Reason 3: distance formula This is already given with the correct reason.
Determine Reason 4 Statement 4: RS ≅ ST . Reason 4: ? Since RS = a and ST = a , we have RS = ST . By the definition of congruence, if two segments have the same length, then they are congruent. Therefore, the reason for statement 4 is the definition of congruence.
Determine Reason 5 Statement 5: RSTU is a square. Reason 5: ? We know that RSTU is a rectangle (given) and that two consecutive sides, RS and ST , are congruent. If two consecutive sides of a rectangle are congruent, then the rectangle is a square. Therefore, the reason for statement 5 is: If two consecutive sides of a rectangle are congruent, then it's a square.
State the Correct Order of Reasons The correct order of reasons is: distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square. This corresponds to option B.
Examples
Understanding geometric proofs, like showing a rectangle is a square, is crucial in architecture and engineering. For example, when designing a building, ensuring that the foundation is perfectly square guarantees stability and equal distribution of weight. This principle applies from the layout of rooms to the construction of windows and doors, where precise shapes are essential for both aesthetics and structural integrity. The properties of squares and rectangles are fundamental in creating balanced and functional designs.
