Answer: Given that MNOP is a trapezium with MN parallel to OP and A is the midpoint of NO.
To prove that the area of triangle MAP is half the area of trapezium MNOP, we need to use the following theorem:
Theorem: If a line segment joins the midpoint of one side of a triangle to a vertex opposite to that side, then the segment divides the triangle into two equal areas.
Proof:
Since A is the midpoint of NO, we can draw a line segment AP from vertex P to point A on NO as shown in the figure below:
M _______N
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O--------P
A
We can see that triangle MAP is formed by the line segment AP and side MP of the trapezium.
Also, we can see that triangle MAN is congruent to triangle NOP because they are both right triangles with corresponding sides parallel. Therefore, they have the same area.
Since MNOP is a trapezium, the area of trapezium is given by:
Area of trapezium MNOP = (1/2)(MN+OP) × height
Since MN is parallel to OP, the height of the trapezium is the same as the height of triangle MAN and NOP.
Therefore, the area of trapezium MNOP can be written as:
Area of trapezium MNOP = (1/2)(MN+OP) × height
= (1/2)(MA+AP+OP) × height
= (1/2)(MA+MP) × height (Since AP = NO/2 = height)
So, we have proved that:
Area of trapezium MNOP = (1/2)(MA+MP) × height
Using the theorem, we know that AP divides triangle MAP into two equal areas. Therefore,
Area of triangle MAP = (1/2) × Area of triangle MAP
Substituting the above in the expression for the area of trapezium, we get:
Area of trapezium MNOP = Area of triangle MAP + (1/2) × Area of triangle MAP
= (3/2) × Area of triangle MAP
Therefore, we have:
Area of triangle MAP = (1/2) × Area of trapezium MNOP
Thus, we have proved that the area of triangle MAP is half the area of trapezium MNOP.