To find the length of the diagonal AC in the kite ABCD, we can use the area formula for a kite, which is given by:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
where \(d_1\) and \(d_2\) are the lengths of the diagonals.
From the problem, we know that the area of the kite is 80 square inches, and we can see that diagonal BE (which is part of diagonal AC) is labeled as 6 inches.
Given the labeling:
- Let \(d_1 = BE\) = 6 inches (the length of diagonal BE)
- Segment AE = 5 inches
- Segment CE = 5 inches
The full length of diagonal AC can be calculated as follows: \[ d_1 = AE + CE = 5 + 5 = 10 , \text{inches} \]
Now, we need to find diagonal BD (which we are not explicitly given). We can denote the lengths of BE and ED as \(h\) (both of which are lengths from point E to points B and D, respectively). Therefore \(d_2\) can be expressed as: \[ d_2 = BE + ED = 6 + h \]
To find the area, we will substitute the known values into the area formula: \[ 80 = \frac{1}{2} \times 10 \times (6 + h) \]
Simplifying the equation: \[ 80 = 5 \times (6 + h) \]
Dividing both sides by 5: \[ 16 = 6 + h \]
Solving for \(h\): \[ h = 16 - 6 = 10 \]
Thus, the full length of diagonal BD is \(BE + ED = 6 + 10 = 16\) inches.
We are ultimately asked for the length of diagonal AC, which we found to be 10 inches.
The answer is: \[ \boxed{10 , \text{in.}} \]