Bd is öne of the diagonal of quad. Abcd.if al | bd and cm| bd ,show that ar(quad.abcd)=1/2xbdx(al+cm)

To prove that the area of quadrilateral ABCD is equal to half of BD times the sum of AL and CM, we can use the concept of triangles and their areas.

First, let's draw the given quadrilateral ABCD and label the points as mentioned:

A_____M_______C
\ | /
\____|____/
B D

Given that AL is parallel to BD and CM is parallel to BD, we can conclude that triangles ABD and BCD are similar. Similarly, triangles ACM and ADM are also similar.

To prove this, we can use the property that if two lines are parallel, the corresponding angles are equal. Therefore, angle ABD = angle BDC and angle ACM = angle ADM.

Now, let's calculate the area of quadrilateral ABCD.

We know that the area of a triangle is given by:

Area = (1/2) * base * height

We can split ABCD into two triangles: ABD and BCD.

1. Area of triangle ABD:
The base is BD, and the height can be calculated as the perpendicular distance between AL and BD. Let's call this distance h1.
Therefore, the area of triangle ABD is (1/2) * BD * h1.

2. Area of triangle BCD:
The base is BD, and the height can be calculated as the perpendicular distance between CM and BD. Let's call this distance h2.
Therefore, the area of triangle BCD is (1/2) * BD * h2.

Adding both areas, we get:
Area of ABCD = (1/2) * BD * h1 + (1/2) * BD * h2

Now, let's substitute h1 and h2:

Since AL is parallel to BD, h1 is equal to the perpendicular distance between AL and CM. Let's call this distance x.
Similarly, CM is parallel to BD, so h2 is equal to the perpendicular distance between AL and CM, which is also x.

Substituting h1 = x and h2 = x, we get:

Area of ABCD = (1/2) * BD * x + (1/2) * BD * x
= (BD * x + BD * x) / 2
= BD * x

Now, let's substitute BD with AL + CM:

Area of ABCD = (AL + CM) * x

Finally, as given in the question, we are asked to prove that the area of ABCD is equal to half of BD times the sum of AL and CM. Substituting the value of Area of ABCD, it becomes:

(AL + CM) * x = (1/2) * BD * (AL + CM)

So, we have successfully proved that the area of quadrilateral ABCD is equal to half of BD times the sum of AL and CM.