Question

Find the midpoint of each diagonal of a square with side of length \(s\). Draw the conclusion that the diagonals of a square intersect at their midpoints. [Hint: Use \((0,0),(0, s),(s, 0),\) and \((s, s)\) as the vertices of the square. \(]\)

Short answer

The diagonals of the square intersect at their midpoints \(( \frac{s}{2}, \frac{s}{2} )\).

Step-by-step solution

- Identify the vertices

Identify the vertices of the square based on the given hint. The vertices are \((0,0)\), \((0, s)\), \((s, 0)\) and \((s, s)\).

- Find the diagonals

Determine the diagonals by connecting opposite vertices. The diagonals of the square are from \((0,0)\) to \((s, s)\) and from \((0, s)\) to \((s, 0)\).

- Compute the midpoint of the first diagonal

The midpoint of the diagonal from \((0,0)\) to \((s, s)\) is calculated as follows: \[ \left( \frac{0 + s}{2}, \frac{0 + s}{2} \right) = \left( \frac{s}{2}, \frac{s}{2} \right) \]

- Compute the midpoint of the second diagonal

The midpoint of the diagonal from \((0, s)\) to \((s, 0)\) is calculated as follows: \[ \left( \frac{0 + s}{2}, \frac{s + 0}{2} \right) = \left( \frac{s}{2}, \frac{s}{2} \right) \]

- Conclusion

Both midpoints are \(( \frac{s}{2}, \frac{s}{2} )\). This indicates that the diagonals of the square intersect at their midpoints.

Key concepts

Midpoint Formula

To understand the midpoint of a line segment, we use the midpoint formula. The formula finds the exact center between two endpoints. If we have two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \(M\) is calculated as: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] This formula takes the average of the x-coordinates and the y-coordinates of the endpoints. This average gives us the middle point.
In the case of our square with vertices \((0,0)\), \((0, s)\), \((s, 0)\), and \((s, s)\), we use this formula to find the midpoint of the diagonals. For instance, diagonal connecting \((0,0)\) to \((s, s)\) yields the midpoint \((\frac{s}{2}, \frac{s}{2})\). Similarly, for the diagonal connecting \((0, s)\) to \((s, 0)\), the midpoint is \((\frac{s}{2}, \frac{s}{2})\).
Applying the midpoint formula correctly helps in verifying that both midpoints are the same. This confirms the intersection point.

Square Properties

A square is a special geometric shape with unique properties. All sides of a square have equal length. If one side measures length \(s\), then all four sides are \(s\). All angles in a square are right angles (90 degrees).
Another fascinating property is that the diagonals of a square are equal in length and they bisect each other at 90 degrees. When we draw the diagonals by connecting opposite vertices, they form two congruent triangles within the square. These triangles help in dividing the square into symmetrical parts.
When computing the midpoint of the diagonals, due to these properties, we notice that both midpoints are identical, highlighting the symmetric nature of the square. This property is essential in many areas of geometry and helps solve various mathematical problems efficiently.

Diagonals Intersection

The point where diagonals intersect is critical for understanding symmetry in a square. As we calculate, both diagonals meet at their midpoint. In our solved example, both diagonals intersect at \((\frac{s}{2}, \frac{s}{2})\).
To determine the point of intersection, we consider each diagonal's midpoint. For diagonal 1, connecting \((0,0)\) to \((s, s)\), the midpoint is \((\frac{s}{2}, \frac{s}{2})\). For diagonal 2, connecting \((0, s)\) to \((s, 0)\), the midpoint is also \((\frac{s}{2}, \frac{s}{2})\).
The intersection at midpoints demonstrates that any square’s diagonals will always meet at the center point. This intersection bisects the diagonals into equal halves, maintaining the properties of a square. Understanding this concept thoroughly helps in visualizing and solving more complex geometric problems.