How do you know that the diagonals of your constructed square areperpendicular bisectors? Explain.

Answer 1

To demonstrate that the diagonals of a constructed square are perpendicular bisectors, we can use both geometric properties and coordinate geometry.

Geometric Properties:

1. Equal Lengths: In a square, all four sides are equal, and all angles are right angles (90 degrees). The diagonals of a square connect opposite corners. In a square of side length \( s \), the length of each diagonal can be calculated using the Pythagorean theorem: \[ d = \sqrt{s^2 + s^2} = s\sqrt{2}. \] Since both diagonals are of equal length (\( d \)), this establishes that they are equal.

2. Angles: The diagonals split the square into four right triangles. Each angle formed by the intersection of the diagonals is made up of two angles from the square (each being a right angle). Therefore, the diagonals intersect at a 90-degree angle, confirming that they are perpendicular to each other.

3. Midpoint: The midpoint of both diagonals is the center of the square. Since the diagonals connect opposite corners, and their intersection point is equidistant from all four vertices, it can be seen that both diagonals bisect each other.

Coordinate Geometry:

Let’s place a square on the coordinate plane for a clearer understanding:

• Consider the square with vertices at \( (0, 0) \), \( (s, 0) \), \( (s, s) \), and \( (0, s) \).

1. Diagonals Definition:

   • The first diagonal connects \( (0, 0) \) and \( (s, s) \).
   • The second diagonal connects \( (0, s) \) and \( (s, 0) \).

2. Finding Slopes:

   • The slope of diagonal 1: \[ \text{slope} = \frac{s - 0}{s - 0} = 1. \]
   • The slope of diagonal 2: \[ \text{slope} = \frac{0 - s}{s - 0} = -1. \]

3. Product of Slopes: If two lines are perpendicular, the product of their slopes should equal \(-1\): \[ 1 \times (-1) = -1. \] This confirms that the diagonals are perpendicular.

4. Midpoint of the Diagonals:

   • The midpoint of diagonal 1 is: \[ \left(\frac{0+s}{2}, \frac{0+s}{2}\right) = \left(\frac{s}{2}, \frac{s}{2}\right). \]
   • The midpoint of diagonal 2 is also: \[ \left(\frac{0+s}{2}, \frac{s+0}{2}\right) = \left(\frac{s}{2}, \frac{s}{2}\right). \] Therefore, both diagonals intersect at the same point, confirming that they bisect each other.

Conclusion:

Thus, through geometric properties as well as coordinate geometry, we have shown that the diagonals of a square are indeed perpendicular bisectors. They are equal in length, intersect at right angles, and bisect each other at their midpoint.

Answer 2

in 5 sentences and on an 8th grade level

Answer 3

The diagonals of a square are the lines that connect opposite corners. They are equal in length and cross each other at the center of the square. Since each corner of the square has right angles, the diagonals meet at a 90-degree angle, meaning they are perpendicular. Furthermore, the point where they intersect is the midpoint of both diagonals, showing that they bisect each other. Therefore, the diagonals of a square are perpendicular bisectors, making them a special feature of squares.

Answer 4

How do you know that the diagonals of your constructed square are congruent? Explain.
on an 8th grade level and in 5 sentences

Answer 5

The diagonals of a square are the lines that connect opposite corners. Since all sides of a square are equal in length, we can use the Pythagorean theorem to find the length of the diagonals. For a square with side length \( s \), each diagonal measures \( s\sqrt{2} \). This means both diagonals have the same length because they are calculated using the same formula. Therefore, the diagonals of a square are congruent, which means they are exactly the same length.