Proving Diagonals Of Square PQRS Are Perpendicular Bisectors

Determining if the diagonals of a square are perpendicular bisectors involves demonstrating specific geometric properties. In the case of square PQRS, the diagonals are line segments that connect opposite vertices, namely PR and QS. To prove that these diagonals are perpendicular bisectors of each other, we need to show two key aspects: first, that the diagonals intersect at a right angle (perpendicularity), and second, that the point of intersection divides each diagonal into two equal segments (bisection).

Side Lengths and Slopes: Establishing the Foundation

Side lengths being equal is a fundamental property of squares. If we establish that the lengths of ${\overline{SP}}$, ${\overline{PQ}}$, ${\overline{RQ}}$, and ${\overline{SR}}$ are each 5, this confirms that PQRS is at least a rhombus, which is a quadrilateral with all four sides of equal length. However, a rhombus does not necessarily have right angles. To confirm that PQRS is indeed a square, we also need to demonstrate that the adjacent sides are perpendicular. This is where the concept of slopes becomes crucial.

If the slope of ${\overline{SP}}$ and ${\overline{RQ}}$ is undefined, it indicates that these sides are vertical lines. This also implies that the sides ${\overline{PQ}}$ and ${\overline{SR}}$ are horizontal lines because they are perpendicular to the vertical sides. The product of the slopes of two perpendicular lines is -1 (except when one is vertical and the other is horizontal). Therefore, demonstrating that adjacent sides have slopes that are negative reciprocals of each other—or that one pair is vertical and the other is horizontal—confirms that the angles at the vertices (P, Q, R, and S) are right angles. Combining equal side lengths and right angles conclusively proves that PQRS is a square.

Diagonals: Perpendicularity and Bisection

Perpendicularity of Diagonals

To prove that the diagonals ${\overline{PR}}$ and ${\overline{QS}}$ are perpendicular, we need to show that their slopes are negative reciprocals of each other. The slope of a line is a measure of its steepness, calculated as the change in the y-coordinate divided by the change in the x-coordinate (rise over run). If the product of the slopes of ${\overline{PR}}$ and ${\overline{QS}}$ is -1, then the diagonals intersect at a 90-degree angle. For example, if the slope of ${\overline{PR}}$ is 2 and the slope of ${\overline{QS}}$ is -1/2, their product is -1, confirming perpendicularity.

Alternatively, if one diagonal is horizontal (slope of 0) and the other is vertical (undefined slope), they are also perpendicular. In a coordinate plane, a horizontal line has a constant y-value, and a vertical line has a constant x-value. Therefore, demonstrating that the slopes of the diagonals are negative reciprocals, or that one is horizontal and the other is vertical, proves the perpendicularity of the diagonals in square PQRS.

Bisection of Diagonals

To prove that the diagonals ${\overline{PR}}$ and ${\overline{QS}}$ bisect each other, we need to show that they intersect at their midpoints. The midpoint of a line segment is the point that divides the segment into two equal parts. The coordinates of the midpoint are calculated by averaging the x-coordinates and the y-coordinates of the endpoints. If the midpoint of ${\overline{PR}}$ is the same as the midpoint of ${\overline{QS}}$, it means the diagonals intersect at their respective midpoints, thus bisecting each other.

For instance, if the coordinates of P are (1, 2), R are (5, 6), Q are (1, 6), and S are (5, 2), the midpoint of ${\overline{PR}}$ is ((1+5)/2, (2+6)/2) = (3, 4), and the midpoint of ${\overline{QS}}$ is ((1+5)/2, (6+2)/2) = (3, 4). Since the midpoints are the same, the diagonals bisect each other. This property, combined with the perpendicularity of the diagonals, definitively proves that ${\overline{PR}}$ and ${\overline{QS}}$ are perpendicular bisectors of each other in square PQRS.

Congruent Triangles: An Alternative Proof

Another approach to proving that the diagonals of square PQRS are perpendicular bisectors involves demonstrating that triangles formed by the diagonals are congruent. If we can show that triangles like ${\triangle POQ}$, ${\triangle QOR}$, ${\triangle ROS}$, and ${\triangle SOP}$ (where O is the intersection point of the diagonals) are congruent, it implies that the diagonals bisect each other and are perpendicular. Congruence can be proven using various postulates, such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS).

In a square, all sides are equal, so PQ = QR = RS = SP. The angles at the vertices are right angles, meaning ${\angle PQR}$ = ${\angle QRS}$ = ${\angle RSP}$ = ${\angle SPQ}$ = 90°. If the diagonals bisect each other, then PO = OR and QO = OS. If the diagonals are perpendicular, then ${\angle POQ}$ = ${\angle QOR}$ = ${\angle ROS}$ = ${\angle SOP}$ = 90°. Using the SAS postulate, if we can show that two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.

For example, consider triangles ${\triangle POQ}$ and ${\triangle QOR}$. If we prove that PO = OR, QO is a common side, and ${\angle POQ}$ = ${\angle QOR}$, then ${\triangle POQ}$ ${\cong}$ ${\triangle QOR}$ by SAS. Similarly, we can prove the congruence of all four triangles. Congruent triangles imply that their corresponding parts are equal. Therefore, if these triangles are congruent, the diagonals bisect each other at right angles, confirming that the diagonals of square PQRS are perpendicular bisectors.

Coordinate Geometry: A Quantitative Approach

Coordinate geometry provides a quantitative approach to proving that the diagonals of square PQRS are perpendicular bisectors. By assigning coordinates to the vertices of the square, we can use algebraic methods to calculate lengths, slopes, and midpoints. This approach offers a concrete way to verify the geometric properties in question.

First, we establish a coordinate system and assign coordinates to the vertices P, Q, R, and S. Without loss of generality, let's assume that P is at the origin (0, 0). Since PQRS is a square, the sides are of equal length and the angles are right angles. If we assume the side length is ${a}$, we can assign coordinates as follows: P(0, 0), Q(a, 0), R(a, a), and S(0, a). These coordinates satisfy the conditions for a square with side length ${a}$.

Next, we calculate the slopes of the diagonals ${\overline{PR}}$ and ${\overline{QS}}$. The slope ${m}$ between two points (x₁, y₁) and (x₂, y₂) is given by ${m = (y₂ - y₁) / (x₂ - x₁)}$. The slope of ${\overline{PR}}$ is ${(a - 0) / (a - 0) = 1}$, and the slope of ${\overline{QS}}$ is ${(a - 0) / (0 - a) = -1}$. The product of the slopes is ${1 imes -1 = -1}$, confirming that the diagonals are perpendicular.

To show bisection, we calculate the midpoints of ${\overline{PR}}$ and ${\overline{QS}}$. The midpoint (xₘ, yₘ) between two points (x₁, y₁) and (x₂, y₂) is given by ${(xₘ, yₘ) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)}$. The midpoint of ${\overline{PR}}$ is ${((0 + a) / 2, (0 + a) / 2) = (a/2, a/2)}$, and the midpoint of ${\overline{QS}}$ is ${((a + 0) / 2, (0 + a) / 2) = (a/2, a/2)}$. Since the midpoints are the same, the diagonals bisect each other. This coordinate geometry approach provides a robust algebraic proof that the diagonals of square PQRS are perpendicular bisectors of each other.

Conclusion

In conclusion, proving that the diagonals of square PQRS are perpendicular bisectors of each other involves demonstrating that they intersect at right angles and that the intersection point divides each diagonal into two equal segments. This can be achieved through various methods, including analyzing side lengths and slopes, proving triangle congruence, and using coordinate geometry. Each approach provides a different perspective on the geometric properties of squares and their diagonals, reinforcing the fundamental principles of Euclidean geometry.