To verify that a constructed shape is truly a square based on the properties of its diagonals, we can address each question in the following manner:
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How do you know that the diagonals of your constructed square are perpendicular bisectors? Explain.
In a square, by definition, the diagonals bisect each other at right angles (90 degrees). This can be demonstrated through the properties of symmetry and the equal length of the sides. Since a square has four equal sides and the angles between those sides are all 90 degrees, when the diagonals are drawn, they create two pairs of congruent triangles. Each diagonal divides the square into two equal halves, and the angle at which they intersect is always 90 degrees (since the opposite angles of a square are always equal). Therefore, each diagonal is indeed a perpendicular bisector of the other, bisecting the angles at the vertices of the square as well.
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How do you know that the diagonals of your constructed (equal lengths) square are congruent? Explain.
The diagonals of a square are congruent due to the properties of the square's geometry. In a square, all four sides are of equal length (let's denote this length as \(s\)). The relationship between the side length \(s\) and the diagonal \(d\) of the square can be derived using the Pythagorean theorem. For a square, the diagonal splits the square into two right triangles, where each leg of the triangle measures \(s\).
According to the Pythagorean theorem, the length of the diagonal \(d\) can be calculated as:
\[ d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2} \]
Since both diagonals are formed by the same sides of the square and have the same calculations applied, they are equal in length. Therefore, the diagonals of a square are congruent.
In summary, a constructed shape qualifies as a square if both properties pertaining to the diagonals—being perpendicular bisectors and being congruent—hold true, as proven by the foundational geometric principles concerning squares.