OIS ANY POINT ON DIAGONAL DB OF PARALLOGRAM ABCD PROVE THAT, AR(TRIANGLE OAB) = AR(TRIANGLE OBC)
To prove that the areas of triangles OAB and OBC are equal in a parallelogram ABCD, let's break down the steps:
Step 1: Understand the problem
We are given a parallelogram ABCD, where O is any point on the diagonal DB. We need to prove that the areas of triangles OAB and OBC are equal.
Step 2: Recall the properties of parallelograms
In a parallelogram, opposite sides are parallel and equal in length. The diagonals also bisect each other—this means they divide each other into two equal parts.
Step 3: Identify relevant triangles
To prove that the areas of triangles OAB and OBC are equal, we need to focus on these two triangles: OAB and OBC.
Step 4: Find the base and height of each triangle
Let's denote the height of both triangles as 'h'. To find the base of triangle OAB, we can use the length of AB, whereas for triangle OBC, we use the length of BC.
Step 5: Establish the relationship between base and height
Since AB and BC are parallel, they have the same length. Therefore, the bases of triangles OAB and OBC are equal.
Step 6: Calculate the area of each triangle
The area of a triangle is given by the formula: area = (1/2) * base * height. Since the bases are equal, we can simplify the formulas as follows:
Area of triangle OAB = (1/2) * AB * h
Area of triangle OBC = (1/2) * BC * h
Step 7: Apply the given information
We know that O is any point on diagonal DB. Since diagonal DB bisects the parallelogram, we can say that AB and BC are equal in length. Therefore, AB = BC.
Step 8: Equate the areas of the triangles
By substituting AB = BC in the area formulas, we get:
Area of triangle OAB = (1/2) * AB * h = (1/2) * BC * h = Area of triangle OBC
Thus, we have proved that AR(triangle OAB) = AR(triangle OBC) in any parallelogram ABCD, where O is any point on the diagonal DB.