ABCD is a parallelogram. Let C′ be a point on AC extended such that the length of AC′=1.2AC. Let D′ be on the segment BD such that the length of BD′=0.9BD. The ratio of the area of the quadrilateral ABC′D′ to the area of the parallelogram ABCD can be written as ab, where a and b are coprime positive integers. What is the value of a+b?

To find the ratio of the area of the quadrilateral ABC′D′ to the area of the parallelogram ABCD, we need to determine the areas of both shapes.

First, let's find the area of parallelogram ABCD. We know that the area of a parallelogram is given by the base multiplied by the height. In this case, we can consider either AB or BC as the base.

Since ABCD is a parallelogram, the height of the parallelogram can be found by drawing a perpendicular line from one of the vertices to the opposite side. In this case, we can draw a perpendicular line from D to BC.

Let's say the height of the parallelogram is h. Since ABCD is a parallelogram, the length of this perpendicular line is the same as the length of BC (or AB). Therefore, the length of this perpendicular line is h.

We are given that length of BD′ is 0.9 times the length of BD. This means that the length of BD′ can be expressed as 0.9h.

Similarly, the length of AC′ is 1.2 times the length of AC. This means that the length of AC′ can be expressed as 1.2h.

Now, let's find the area of the quadrilateral ABC′D′. To do this, we need to find the lengths of the bases and the height of the quadrilateral.

The length of the base on which we can consider the quadrilateral is BC, which is h.

The length of the other base, on which we cannot consider the quadrilateral, is AC. However, since C′ is a point on AC extended, the distance between C and C′ is AC′ - AC = 1.2h - h = 0.2h.

So, the length of the other base of the quadrilateral ABC′D′ is 0.2h.

To find the height of the quadrilateral, we need to calculate the vertical distance between the lines determined by BD and AC′. Since C′ is on AC extended, this vertical distance is given by 0.9h.

Now, we can calculate the area of the quadrilateral ABC′D′ as (0.2h) * (0.9h) = 0.18h^2.

The ratio of the area of the quadrilateral ABC′D′ to the area of the parallelogram ABCD is (0.18h^2) / (h * h) = 0.18.

Therefore, the ratio of the areas is 0.18, and the value of a + b is 0 + 18 = 18.

This is a question of Brilliant[dot]org

jishka please ban the IP address of this user. He has posted Brilliant problems earlier,too.