Which Statement Proves That The Diagonals Of Square { PQRS $}$ Are Perpendicular Bisectors Of Each Other?A. The Midpoint Of Both Diagonals Is { ( 4 \frac{1}{2}, 5 \frac{1}{2} )$}$, The Slope Of { \overline{RP}$}$ Is 7,
Introduction
In geometry, a square is a special type of quadrilateral where all four sides are of equal length, and all internal angles are right angles. One of the key properties of a square is that its diagonals are perpendicular bisectors of each other. This means that the diagonals intersect at their midpoints and form right angles. In this article, we will explore the properties of diagonals in a square and determine which statement proves that the diagonals are perpendicular bisectors of each other.
Properties of Diagonals in a Square
A diagonal of a square is a line segment that connects two opposite vertices of the square. Since the diagonals of a square are perpendicular bisectors of each other, they divide the square into four congruent right triangles. Each right triangle has a hypotenuse that is a side of the square and two legs that are halves of the diagonals.
Midpoint of Diagonals
The midpoint of a diagonal is the point where the diagonal intersects the midpoint of the opposite side. In a square, the midpoint of both diagonals is the same point. This point is also the center of the square.
Slope of Diagonals
The slope of a diagonal is the ratio of the vertical change to the horizontal change between two points on the diagonal. Since the diagonals of a square are perpendicular bisectors of each other, their slopes are negative reciprocals of each other.
Statement Analysis
Let's analyze the given statement:
A. The midpoint of both diagonals is {( 4 \frac{1}{2}, 5 \frac{1}{2} )$}$, the slope of {\overline{RP}$}$ is 7.
To determine if this statement proves that the diagonals are perpendicular bisectors of each other, we need to examine the midpoint and slope of the diagonals.
Midpoint Analysis
The midpoint of both diagonals is given as {( 4 \frac{1}{2}, 5 \frac{1}{2} )$}$. This means that the diagonals intersect at this point, which is the center of the square. Since the diagonals are perpendicular bisectors of each other, their midpoints must coincide.
Slope Analysis
The slope of {\overline{RP}$}$ is given as 7. Since the diagonals of a square are perpendicular bisectors of each other, their slopes are negative reciprocals of each other. Therefore, the slope of the other diagonal must be -1/7.
Conclusion
Based on the analysis of the midpoint and slope of the diagonals, we can conclude that the given statement proves that the diagonals of square { PQRS $}$ are perpendicular bisectors of each other.
Proof
To prove that the diagonals are perpendicular bisectors of each other, we need to show that the midpoint of both diagonals is the same point and that their slopes are negative reciprocals of each other.
Let's consider the diagonals {\overline{PR}$}$ and {\overline{QS}$}$. Since the diagonals are perpendicular bisectors of each other, their midpoints coincide. Let's call this midpoint {M$}$.
The midpoint of {\overline{PR}$}$ is given by:
{M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$]
where [(x_2, y_2)$] are the coordinates of points [P\$} and {R$}$, respectively.
Similarly, the midpoint of {\overline{QS}$}$ is given by:
{M = \left( \frac{x_3 + x_4}{2}, \frac{y_3 + y_4}{2} \right)$]
where [(x_4, y_4)$] are the coordinates of points [Q\$} and {S$}$, respectively.
Since the midpoints of both diagonals are the same point, we can equate the two expressions:
{\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{x_3 + x_4}{2}, \frac{y_3 + y_4}{2} \right)$]
Simplifying the equation, we get:
[x_1 + x_2 = x_3 + x_4\$}
{y_1 + y_2 = y_3 + y_4$}$
This shows that the midpoint of both diagonals is the same point, which is the center of the square.
Now, let's consider the slopes of the diagonals. The slope of {\overline{PR}$}$ is given by:
{m_1 = \frac{y_2 - y_1}{x_2 - x_1}$]
Similarly, the slope of [\overline{QS}\$} is given by:
{m_2 = \frac{y_4 - y_3}{x_4 - x_3}$]
Since the diagonals are perpendicular bisectors of each other, their slopes are negative reciprocals of each other. Therefore, we can write:
[m_1 \cdot m_2 = -1\$}
Substituting the expressions for {m_1$}$ and {m_2$}$, we get:
{\frac{y_2 - y_1}{x_2 - x_1} \cdot \frac{y_4 - y_3}{x_4 - x_3} = -1$}$
Simplifying the equation, we get:
{(y_2 - y_1)(y_4 - y_3) = -(x_2 - x_1)(x_4 - x_3)$}$
This shows that the slopes of the diagonals are negative reciprocals of each other, which proves that the diagonals are perpendicular bisectors of each other.
Conclusion
Q: What is the property of diagonals in a square?
A: The diagonals of a square are perpendicular bisectors of each other. This means that the diagonals intersect at their midpoints and form right angles.
Q: What is the midpoint of a diagonal?
A: The midpoint of a diagonal is the point where the diagonal intersects the midpoint of the opposite side. In a square, the midpoint of both diagonals is the same point, which is also the center of the square.
Q: What is the slope of a diagonal?
A: The slope of a diagonal is the ratio of the vertical change to the horizontal change between two points on the diagonal. Since the diagonals of a square are perpendicular bisectors of each other, their slopes are negative reciprocals of each other.
Q: How do you prove that the diagonals of a square are perpendicular bisectors of each other?
A: To prove that the diagonals of a square are perpendicular bisectors of each other, you need to show that the midpoint of both diagonals is the same point and that their slopes are negative reciprocals of each other. This can be done by using the midpoint formula and the slope formula.
Q: What is the significance of the diagonals of a square being perpendicular bisectors of each other?
A: The diagonals of a square being perpendicular bisectors of each other is a fundamental property of a square. It has many practical applications in geometry, trigonometry, and other branches of mathematics.
Q: Can the diagonals of a square be parallel?
A: No, the diagonals of a square cannot be parallel. Since the diagonals are perpendicular bisectors of each other, they must intersect at their midpoints, which means they cannot be parallel.
Q: Can the diagonals of a square be congruent?
A: Yes, the diagonals of a square are congruent. Since the diagonals are perpendicular bisectors of each other, they must have the same length.
Q: What is the relationship between the diagonals of a square and its sides?
A: The diagonals of a square are related to its sides in that they divide the square into four congruent right triangles. Each right triangle has a hypotenuse that is a side of the square and two legs that are halves of the diagonals.
Q: Can the diagonals of a square be used to find the area of the square?
A: Yes, the diagonals of a square can be used to find the area of the square. Since the diagonals are perpendicular bisectors of each other, they divide the square into four congruent right triangles. The area of each right triangle can be found using the formula for the area of a triangle, and then the total area of the square can be found by multiplying the area of one right triangle by 4.
Q: Can the diagonals of a square be used to find the perimeter of the square?
A: Yes, the diagonals of a square can be used to find the perimeter of the square Since the diagonals are perpendicular bisectors of each other, they divide the square into four congruent right triangles. The length of each side of the square can be found using the Pythagorean theorem, and then the perimeter of the square can be found by multiplying the length of one side by 4.
Conclusion
In conclusion, the diagonals of a square are perpendicular bisectors of each other, and this property has many practical applications in geometry, trigonometry, and other branches of mathematics. The diagonals can be used to find the area and perimeter of the square, and they can also be used to prove that the square is a special type of quadrilateral.