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 - property

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, \(DO=OB = 20\) inches.

Step2: Use the Pythagorean theorem in \(\triangle AOD\)

In right - triangle \(\triangle AOD\), \(AD = 29\) inches and \(DO = 20\) inches. By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = AD\), \(a = DO\), and \(b = AO\). So, \(AO=\sqrt{AD^{2}-DO^{2}}\). Substitute \(AD = 29\) and \(DO = 20\) into the formula: \(AO=\sqrt{29^{2}-20^{2}}=\sqrt{(29 + 20)(29 - 20)}=\sqrt{49\times9}=\sqrt{441}=21\) inches.

Step3: Use the Pythagorean theorem in \(\triangle DOC\)

In right - triangle \(\triangle DOC\), \(DC = 25\) inches and \(DO = 20\) inches. By the Pythagorean theorem, \(CO=\sqrt{DC^{2}-DO^{2}}\). Substitute \(DC = 25\) and \(DO = 20\) into the formula: \(CO=\sqrt{25^{2}-20^{2}}=\sqrt{(25 + 20)(25 - 20)}=\sqrt{45\times5}=\sqrt{225}=15\) inches.

Step4: Calculate the length of \(AC\)

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

Answer:

36