Final answer:
The intersection point of the diagonals of square LMNO is (0, 0), which shows that the diagonals are congruent perpendicular bisectors of each other, confirming that option A is correct.
Step-by-step explanation:
The question asks which of the given points shows that the diagonals of square LMNO are congruent perpendicular bisectors of each other. To prove this, we must show that the diagonals are equal in length, intersect at a 90° angle, and bisect each other, which means they intersect at their midpoint.
To find the intersection point of the diagonals, we'll look at the coordinates of the vertices. Vertices L and N are opposite each other, as are vertices M and O. If we draw straight lines from L to N and from M to O, these lines would represent the diagonals of the square. The midpoint of each diagonal, where they would intersect, can be found by averaging the x-coordinates and the y-coordinates of the endpoints of each diagonal separately:
- Midpoint of LN: ((0+0)/2, (2+(-2))/2) = (0/2, 0/2) = (0,0)
- Midpoint of MO: ((2+(-2))/2, (0+0)/2) = (0/2, 0/2) = (0,0)
Both midpoints result in the point (0, 0), which is the center of the square and the point of intersection for the diagonals, demonstrating that they bisect each other. To prove that the diagonals are congruent, we can use the distance formula:
= √((x2 - x1)² + (y2 - y1)²)
The length of diagonal LN is equal to the length of diagonal MO since both connect opposite vertices of a square. The diagonals are also perpendicular to each other as they intersect at the right angles typical of square diagonals. Hence, with (0, 0) as the intersection point, the diagonals are congruent perpendicular bisectors.
Therefore, option A) (0, 0) is correct.