State and prove the converse of the following theorem. “If a line divides any two sides of a triangle in the same ratio it must be parallel to the third side”. Using the result, prove the following:
"A line drawn from the midpoint of a non-parallel side of a trapezium, parallel to the parallel sides, bisects the other non-parallel side.
Step 1: Note the given data and draw a diagram
Let be a triangle.
In , the line intersects and in and respectively.
Given that ……(i)
Let the line is not parallel to .
Construction: Draw a line ( assume that is parallel to )
Step 2: Proving the basic proportionality theorem :
In
Since , so according to basic proportionality theorem
Since,
we know that, the corresponding sides of similar triangles are proportional.
…..(ii)
Equating equation (i) and equation (ii)
Adding both sides
This is only possible when is coincide with .
Thus is parallel to .
Hence proved.
Step 3: Construct a trapezium and note the given data
Let be a trapezium with .
let, are midpoints of , draw a line from meeting at . Such that, .
Given, .
The diagram is shown below.
Step 4: Applying Thales theorem in and
Thales theorem states that if a line drawn parallel to one side of a triangle intersects the other two sides in distinct points, the other two sides are divided in the same ratio.
Here
Here
Equating equations (i) and (ii)
Hence, the statement “A line drawn from the midpoint of a non-parallel side of a trapezium, parallel to the parallel sides, bisects the other non-parallel side” is proved