Which statement proves that the diagonals of square PQRS are perpendicular bisectors of each other?
The length of $\overline{ SP }, \overline{ PQ }, \overline{ RQ }$, and $\overline{ SR }$ are each 5 .
The slope of $\overline{ SP }$ and $\overline{ RQ }$ is $-\frac{4}{3}$ and the slope of $\overline{ SR }$ and $\overline{ PQ }$ is $\frac{3}{4}$.
The length of $\overline{ SQ }$ and $\overline{ RP }$ are both $\sqrt{50}$.
The midpoint of both diagonals is $\left(4 \frac{1}{2}, 5 \frac{1}{2}\right)$, the slope of $\overline{ RP }$ is 7 , and the slope of $\overline{ SQ }$ is $-\frac{1}{7}$.
Answer (1)
Statement 1 shows all sides are equal, but it doesn't prove the angles are 90 degrees.
Statement 2 shows opposite sides are parallel and adjacent sides are perpendicular, indicating a rectangle.
Statement 3 shows the diagonals are equal, confirming it's a rectangle.
Statement 4 shows the diagonals bisect each other and are perpendicular.
Therefore, statement 4 proves the diagonals are perpendicular bisectors of each other: Statement 4
Explanation
Problem Analysis Let's analyze each statement to determine which one proves that the diagonals of square PQRS are perpendicular bisectors of each other. A square has the properties that all sides are equal, all angles are 90 degrees, and the diagonals are equal, bisect each other, and are perpendicular.
Analyzing Statement 1 Statement 1: The length of SP , PQ , RQ , and SR are each 5. This tells us that all sides are equal. However, this alone doesn't guarantee that the angles are 90 degrees. The shape could be a rhombus, which has equal sides but not necessarily right angles. In a rhombus, the diagonals are perpendicular bisectors, but this statement alone is not sufficient to prove that PQRS is a square.
Analyzing Statement 2 Statement 2: The slope of SP and RQ is − 3 4 and the slope of SR and PQ is 4 3 . This indicates that opposite sides are parallel, since they have the same slope. Furthermore, the product of the slopes of adjacent sides (e.g., SP and SR ) is − 3 4 ⋅ 4 3 = − 1 . This means that adjacent sides are perpendicular, so all angles are 90 degrees. This tells us that PQRS is a rectangle.
Analyzing Statement 3 Statement 3: The length of SQ and RP are both 50 . This means that the diagonals are equal in length. In a parallelogram, if the diagonals are equal, then the parallelogram is a rectangle.
Analyzing Statement 4 Statement 4: The midpoint of both diagonals is ( 2 9 , 2 11 ) , the slope of RP is 7, and the slope of SQ is − 7 1 . The fact that the midpoint of both diagonals is the same means that the diagonals bisect each other. The product of the slopes of the diagonals is 7 ⋅ ( − 7 1 ) = − 1 , which means that the diagonals are perpendicular. Therefore, this statement proves that the diagonals are perpendicular bisectors of each other.
Conclusion Therefore, the fourth statement proves that the diagonals of square PQRS are perpendicular bisectors of each other.
Examples
In architecture, knowing that the diagonals of a square are perpendicular bisectors is crucial for ensuring structural integrity and precise alignment in square-shaped building foundations or room layouts. This property guarantees that the support beams placed along the diagonals meet at a perfect right angle in the center, equally distributing the load and preventing uneven stress distribution, which is vital for the building's stability and longevity.
