Find the missing values by solving the parallelogram shown in the figure.

Parallelograms are very interesting shapes. They are a type of quadrilateral that has two pairs of parallel lines. They find their applications in multiple disciplines like engineering and science. They are also used for many practical purposes. Even rectangles are a type of parallelogram.

Answer: The missing values in the given figure are x = 5 and y = 8.

Let's understand how we arrived at the solution.

Explanation:

Step 1: First of all we check for similarity of triangles AOB and COD.

⇒angle OAB = angle OCD (alternate angles)

⇒angle ABO = angle ODC (alternate angles)

⇒angle AOB = angle COD (vertically opposite angles)

Hence, the triangles AOB and COD are similar by AAA similarity.

By this, we can conclude that AO / CO = BO / DO.

Hence, we see that:

⇒(5 + 2x) / 3x = (x + y) / 13

Also, it is known that AO = CO and BO = DO (diagonals of parallelogram divide each other at the midpoint).

Hence, we get:

⇒ x + y =13 ---- (1)

⇒ 5 + 2x = 3x ---- (2)

We can use the properties of a parallelogram in which, the diagonals bisect each other. We may get the same equations as above.

Hence, solving these, we find that the missing values in the given figure are x = 5 and y = 8.