Question

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? inches

Explanation:

Step1: Recall kite - properties

The diagonals of a kite are perpendicular, and one diagonal is bisected by the other. Let the intersection of the diagonals \(AC\) and \(DB\) be point \(O\). Since \(DB = 40\) inches, let \(DO=x\) and \(OB = 40 - x\). Assume \(DB\) is bisected by \(AC\), so \(DO=OB = 20\) inches.

Step2: Use the Pythagorean theorem in right - triangles

In right - triangle \(AOD\), by the Pythagorean theorem \(AO=\sqrt{AD^{2}-DO^{2}}\). Given \(AD = 29\) inches and \(DO = 20\) inches, then \(AO=\sqrt{29^{2}-20^{2}}=\sqrt{(29 + 20)(29 - 20)}=\sqrt{49\times9}=\sqrt{441}=21\) inches.
In right - triangle \(DOC\), by the Pythagorean theorem \(OC=\sqrt{DC^{2}-DO^{2}}\). Given \(DC = 25\) inches and \(DO = 20\) inches, then \(OC=\sqrt{25^{2}-20^{2}}=\sqrt{(25 + 20)(25 - 20)}=\sqrt{45\times5}=\sqrt{225}=15\) inches.

Step3: Calculate the length of \(AC\)

Since \(AC=AO + OC\), then \(AC=21+15 = 36\) inches.

Answer:

36