Question

In the quadrilateral KMNP, diagonals KM and MP intersect at point O. MO = OP = 6cm ON = 8cm, NP = 10cm Point A does not belong to the plane of the quadrilateral, ΑΝ is perpendicular (MNP). Prove that MP is perpendicular (AON)

Answer 1

To prove that MP is perpendicular to AON in quadrilateral KMNP, we can make use of the fact that the diagonals of a quadrilateral bisect each other. Additionally, we can use the perpendicular bisector theorem to show that AO is perpendicular to NP, which means MP is perpendicular to AO. Therefore, MP is perpendicular to AON.

Step-by-step explanation:

Given that in quadrilateral KMNP, diagonals KM and MP intersect at point O, and MO = OP = 6cm, ON = 8cm, and NP = 10cm, we need to prove that MP is perpendicular to AON.

To prove this, we can make use of the fact that in a quadrilateral, the diagonals bisect each other. Since MO = OP, this implies that point O is the midpoint of the diagonal KM. Similarly, since ON = NP, point O is the midpoint of the diagonal NP.

Now, let's consider the triangle AON. We know that AN is perpendicular to NP. Since O is the midpoint of NP, the line segment AO is the perpendicular bisector of NP. This means that AO is perpendicular to NP.

Since MO = OP, and AO is perpendicular to NP, this implies that MP is also perpendicular to AO. Therefore, we have proved that MP is perpendicular to AON.