Given Information:
- Square JKLM
- P and Q are midpoints of JK and LM respectively
- O is the point of intersection of diagonals JL and KM
- N is the point of intersection of JO and PM
- Area of triangle PON = 1

To Find:
- Area of triangle MON

Solution:

Step 1: Understanding the Given Information
- We have a square JKLM, with P and Q as midpoints of JK and LM respectively.
- O is the point of intersection of diagonals JL and KM.
- N is the point of intersection of JO and PM.

Step 2: Visualizing the Problem
- Let's visualize the given information to get a better understanding of the problem.

Step 3: Analyzing the Triangles
- We have two triangles in the problem - triangle PON and triangle MON.
- We are given that the area of triangle PON is 1.

Step 4: Understanding Triangle PON
- Triangle PON is formed by the points P, O, and N.
- We are given that the area of triangle PON is 1.

Step 5: Properties of Midpoints
- Since P and Q are midpoints of JK and LM respectively, we know that PQ is parallel to JL and PQ = 1/2 JL.
- Similarly, we can conclude that QM is parallel to KM and QM = 1/2 KM.

Step 6: Properties of Diagonals
- Since O is the point of intersection of JL and KM, we can conclude that JO is an angle bisector of angle JKM.
- Similarly, since N is the point of intersection of JO and PM, we can conclude that NO is an angle bisector of angle POM.

Step 7: Using the Properties
- We know that JO is an angle bisector of angle JKM.
- Therefore, angle JON = angle KON.
- Similarly, angle MON = angle NOP.

Step 8: Using Area of Triangle PON
- Let's assume the area of triangle MON as x.
- Since the area of triangle PON is given as 1, we can write:
- Area of triangle PON = 1/2 * base * height = 1
- Base of triangle PON = NO = x
- Height of triangle PON = PQ = 1/2 JL
- Therefore, we have: (1/2) * x * (1/2 JL) = 1
- Simplifying the equation, we get: x * JL = 4

Step 9: Using the Properties of Midpoints
- Since P and Q are midpoints of JK and LM respectively, we know that JL = 2 PQ.
- Substituting the value of JL in the equation from Step 8, we get: x * 2 PQ = 4
- Simplifying the equation, we get: x * PQ = 2

Step 10: Using the Properties of Midpoints (Continued)
- Since P and Q are midpoints of JK and LM respectively, we know that PQ = 1/2 JK