Question

solving problems with kites
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 \\( \overline { d b } \\) has a length of 40 inches. what is the length of the diagonal \\( \overline { a c } \\)?

Explanation:

Step1: Divide the diagonal \(DB\)

Since the diagonals of a kite are perpendicular and one diagonal bisects the other. Given \(DB = 40\) inches, then \(DE=\frac{DB}{2}=20\) inches and \(EB = 20\) inches.

Step2: Use the Pythagorean theorem for \(\triangle ADE\)

In right - triangle \(ADE\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 29\) inches (side \(AD\)) and \(b = 20\) inches (\(DE\)). Let the length of \(AE\) be \(x\). Then \(x=\sqrt{29^{2}-20^{2}}=\sqrt{(29 + 20)(29 - 20)}=\sqrt{49\times9}=\sqrt{441}=21\) inches.

Step3: Use the Pythagorean theorem for \(\triangle CDE\)

In right - triangle \(CDE\), \(c = 25\) inches (side \(CD\)) and \(b = 20\) inches (\(DE\)). Let the length of \(CE\) be \(y\). Then \(y=\sqrt{25^{2}-20^{2}}=\sqrt{(25 + 20)(25 - 20)}=\sqrt{45\times5}=\sqrt{225}=15\) inches.

Step4: Calculate the length of \(AC\)

Since \(AC=AE + EC\), and \(AE = 21\) inches, \(EC = 15\) inches. Then \(AC=21 + 15=36\) inches.

Answer:

\(36\)