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, $overline{eg}$, measures 24 cm. what is the length of the other diagonal, $overline{df}$? $\bigcirc$ 5 cm $\bigcirc$ 16 cm $\bigcirc$ 21 cm $\bigcirc$ 32 cm

Explanation:

Step1: Recall kite diagonal properties

In a kite, one diagonal is the perpendicular bisector of the other. So, diagonal \( EG \) is bisected by \( DF \), meaning \( ED = DG=\frac{EG}{2}\).
Given \( EG = 24\) cm, then \( ED=\frac{24}{2}=12\) cm.

Step2: Use Pythagorean theorem for triangle \( EDF \)

For triangle \( EDF \), \( EF = 20\) cm (side of the kite), \( ED = 12\) cm (half of \( EG \)), and \( DF \) is composed of \( FD_1\) (let's say the upper part) and \( D_1D\) (but actually, we can find the length from \( F \) to \( D \) first and then consider the lower part? Wait, no, actually, the kite has two triangles: upper triangle \( EFG \) and lower triangle \( EDG \)? Wait, no, the diagonals intersect at \( D \)? Wait, the diagram: \( F \) is top, \( D \) is bottom, \( E \) and \( G \) are left and right. So diagonals \( EG \) and \( DF \) intersect at, let's say, point \( O \), which bisects \( EG \) and is perpendicular to \( EG \). So for triangle \( EOF \), \( EF = 20\) cm, \( EO=\frac{EG}{2}=12\) cm. Then by Pythagoras, \( FO=\sqrt{EF^{2}-EO^{2}}=\sqrt{20^{2}-12^{2}}=\sqrt{400 - 144}=\sqrt{256}=16\) cm. Then for the lower triangle \( EOD \), \( ED = 13\) cm, \( EO = 12\) cm, so \( OD=\sqrt{ED^{2}-EO^{2}}=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5\) cm. Then \( DF=FO + OD=16 + 5=21\) cm? Wait, no, wait: Wait, the sides: top two sides \( EF = FG = 20\) cm, bottom two sides \( ED = DG = 13\) cm. So the diagonal \( DF \) is split into two parts by the intersection with \( EG \), say at point \( O \). So \( EO = OG = 12\) cm (since \( EG = 24\)). Then in triangle \( EOF \): \( EF = 20\), \( EO = 12\), so \( FO=\sqrt{20^{2}-12^{2}} = 16\). In triangle \( EOD \): \( ED = 13\), \( EO = 12\), so \( OD=\sqrt{13^{2}-12^{2}} = 5\). Then \( DF = FO + OD = 16 + 5 = 21\) cm. Wait, but let's check again. Wait, maybe I mixed up the triangles. Wait, the kite has two pairs of adjacent sides equal: \( EF = FG = 20\), \( ED = DG = 13\). The diagonals intersect at right angles, and one diagonal is bisected. So \( EG \) is bisected by \( DF \), so \( EO = OG = 12\). Then for the triangle with sides 20, 12, and the segment from \( F \) to \( O \): \( FO=\sqrt{20^2 - 12^2}=16\). For the triangle with sides 13, 12, and the segment from \( D \) to \( O \): \( DO=\sqrt{13^2 - 12^2}=5\). Then \( DF = FO + DO = 16 + 5 = 21\) cm.

Answer:

21 cm (corresponding to the option "21 cm")