Question

sofia cuts a piece of felt in the shape of a kite for an art project. the top two sides measure 20 cm each and the bottom two sides measure 13 cm each. one diagonal, eg, measures 24 cm. what is the length of the other diagonal, df? 5 cm 16 cm 21 cm 32 cm

Explanation:

Step1: Recall kite - diagonal property

The diagonals of a kite are perpendicular, and one diagonal bisects the other diagonal. Let the intersection of the diagonals \(EG\) and \(DF\) be point \(O\). Given \(EG = 24\) cm, so \(EO=OG = 12\) cm.

Step2: Use the Pythagorean theorem in right - triangle \(EOD\)

In right - triangle \(EOD\), \(ED = 13\) cm and \(EO = 12\) cm. By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = ED\), \(a = EO\), and \(b = OD\). Then \(OD=\sqrt{ED^{2}-EO^{2}}\). Substitute \(ED = 13\) cm and \(EO = 12\) cm into the formula: \(OD=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5\) cm.

Step3: Use the Pythagorean theorem in right - triangle \(EOF\)

In right - triangle \(EOF\), \(EF = 20\) cm and \(EO = 12\) cm. By the Pythagorean theorem, \(OF=\sqrt{EF^{2}-EO^{2}}\). Substitute \(EF = 20\) cm and \(EO = 12\) cm into the formula: \(OF=\sqrt{20^{2}-12^{2}}=\sqrt{400 - 144}=\sqrt{256}=16\) cm.

Step4: Calculate the length of \(DF\)

Since \(DF=OD + OF\), and \(OD = 5\) cm, \(OF = 16\) cm, then \(DF=5 + 16=21\) cm.

Answer:

21 cm