First find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals. 

To find the missing diagonal, apply the area formula: 




This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal. 










Therefore, the sum of the two diagonals is: 

This problem can be solved by applying the area formula: 



Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal. 

Thus the solution is: 

We can find the longer diagonal by adding together the altitude of the top triangle and the altitude of the bottom triangle. To find these, use Pythagorean Theorem. We can use Pythagorean Theorem because one of the properties of a kite is that the two diagonals are perpendicular.

The top triangle has two sides of length 3 [labeled in the picture], and a base of 4 [provided in the written directions]. To figure out the altitude, split this triangle into 2 right triangles. The two legs are x [the altitude] and 2 [half of the base 4], and the hypotenuse is 3:

subtract 4 from both sides

take the square root of both sides

We will do something similar for the bottom triangle. Consider one of the right triangles. It will have a hypotenuse of 7, one leg that we don't know, x [the altitude], and one leg 2 [half the shorter diagonal]. Set up the equation using the Pythagorean Theorem:

subtract 4 from both sides

take the square root of both sides

That can be simplified by considering 45 as the product of . Since the square root of 9 is 3, we can re-write as .

Adding together the first answer of plus gives .

1) The two diagonals are  and .

2) To find , use the Law of Cosines and either  or , because either angle is opposite .

3) Using ,  , and the Law of Cosines is

3) To find , use the Law of Cosines using either  or , because either angle is opposite  .

4) Using ,  , and the Law of Cosines is

Let  be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is  units long.

Let  be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is  units long.

Let  be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is  units long.

Let  be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is  units long.

Let  be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is  units long.

Let  be the length of the shorter diagonal. Then the length of the longer diagonal can be represented by .

Recall how to find the area of a kite:

Plug in the given area and solve for .

Since we are dealing with geometric shapes, the answer must be a positive value. Thus, .

The length of the shorter diagonal is  units long.