Introduction:
A quadrilateral is a polygon with four sides and four vertices. When a diagonal is drawn within a quadrilateral, it divides the quadrilateral into two unique triangles. The question is whether the sum of the perimeter of these two triangles will be equal to the perimeter of the original quadrilateral.

Explanation:
To understand whether the sum of the perimeter of the triangles formed by the diagonal is equal to the perimeter of the quadrilateral, let's consider a quadrilateral ABCD with diagonal AC.

Perimeter of Quadrilateral ABCD:
The perimeter of a polygon is the sum of the lengths of all its sides. So, the perimeter of quadrilateral ABCD can be calculated by adding the lengths of its four sides AB, BC, CD, and DA.

Perimeter of Quadrilateral ABCD = AB + BC + CD + DA

Perimeter of Triangles ACD and ABC:
When the diagonal AC is drawn within quadrilateral ABCD, it divides the quadrilateral into two triangles, triangle ACD and triangle ABC.

The perimeter of a triangle is the sum of the lengths of its three sides. So, the perimeter of triangle ACD can be calculated by adding the lengths of its three sides AC, CD, and DA. Similarly, the perimeter of triangle ABC can be calculated by adding the lengths of its three sides AB, BC, and AC.

Perimeter of Triangle ACD = AC + CD + DA
Perimeter of Triangle ABC = AB + BC + AC

Sum of Perimeters of Triangles ACD and ABC:
Now, let's calculate the sum of the perimeters of triangles ACD and ABC.

Sum of Perimeters of Triangles ACD and ABC = (AC + CD + DA) + (AB + BC + AC)

Rearranging the terms, we get:

Sum of Perimeters of Triangles ACD and ABC = (AB + BC + CD + DA) + (AC + AC)

Considering that AC is a common side between the two triangles, we can simplify the equation further:

Sum of Perimeters of Triangles ACD and ABC = (AB + BC + CD + DA) + 2AC

Conclusion:
From the above equation, we can observe that the sum of the perimeters of triangles ACD and ABC is equal to the perimeter of quadrilateral ABCD plus twice the length of the diagonal AC.

Therefore, the sum of the perimeter of the triangles formed by the diagonal is not equal to the perimeter of the original quadrilateral. It is greater than the perimeter of the quadrilateral by twice the length of the diagonal.