the point of intersection of the diagonals of a quadrilateral divides one diagonal in the ratio 2:3. is it parallelogram ? give reasons
To determine whether the given quadrilateral is a parallelogram or not, we need to consider the properties of a parallelogram.
A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. Additionally, the diagonals of a parallelogram bisect each other.
Now, let's analyze the given information.
We are told that the point of intersection of the diagonals of the quadrilateral divides one diagonal in the ratio 2:3. Let's assume the diagonals intersect at point O and label the quadrilateral's vertices as A, B, C, and D.
Let AC be one of the diagonals, and let OD be the divided diagonal, where O is the point of intersection. According to the given ratio, AO:OC is 2:3.
To determine whether the quadrilateral is a parallelogram, we need to prove that opposite sides are parallel. Let's consider the opposite sides of the quadrilateral: AB and CD.
Now, we need to use the concept of similar triangles to analyze the situation.
Since OD divided AC in the ratio 2:3, we can consider triangles OAD and OCD. These two triangles share an angle at O.
According to the Angle-Angle (AA) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Since ∠OAD = ∠OCD (they are vertically opposite angles) and ∠ADO = ∠CDO (they are corresponding angles), triangles OAD and OCD are similar.
Now, let's consider the ratios of the corresponding sides of these similar triangles. We have:
AD/CD = AO/OC -- Ratio of corresponding sides of similar triangles
Since AO/OC = 2/3, we can rewrite the above equation as:
AD/CD = 2/3
This shows that the ratio of the lengths of AD and CD is also 2:3. Therefore, it is clear that opposite sides of the quadrilateral are parallel.
Hence, based on the given information, we can conclude that the quadrilateral formed by the diagonals' point of intersection is a parallelogram.