Kara is building a sandbox shaped like a kite for her nephew. The top two sides of the sandbox are 29 inches long. The bottom two sides are 25 inches long. The diagonal DB has a length of 40 inches. What is the length of the diagonal AC?
In a kite ABCD, the diagonal connecting the vertices where equal sides meet (AC) is the perpendicular bisector of the other diagonal (DB). Let M be the intersection point of the diagonals. Given that the length of the diagonal DB is 40 inches, the segment MB is half of DB, so MB = 20 inches. The triangles AMB and CMB are right-angled triangles at M. Using the Pythagorean theorem for triangle AMB, we have \(AM^2 + MB^2 = AB^2\). Substituting the given values: \(AM^2 + 20^2 = 29^2\). This simplifies to \(AM^2 + 400 = 841\), so \(AM^2 = 441\), which means \(AM = 21\) inches.
Using the Pythagorean theorem for triangle CMB, we have \(CM^2 + MB^2 = BC^2\). Substituting the given values: \(CM^2 + 20^2 = 25^2\). This simplifies to \(CM^2 + 400 = 625\), so \(CM^2 = 225\), which means \(CM = 15\) inches.
The total length of the diagonal AC is the sum of the segments AM and CM. Thus, \(AC = AM + CM = 21 + 15 = 36\) inches. The final length of the diagonal AC is \(\boxed{36}\) inches.
A landscape architect uses this principle when designing geometric garden layouts, such as a kite-shaped flowerbed. By calculating the diagonals of the kite, they can precisely determine the location of central irrigation lines that follow the path of the diagonals, ensuring the garden is both geometrically balanced and efficiently watered. Understanding the relationships between side lengths and diagonals of a kite is a fundamental skill for applying geometry to real-world structural planning.