{"id":962,"date":"2020-10-16T15:59:00","date_gmt":"2020-10-16T19:59:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=962"},"modified":"2024-10-25T11:47:21","modified_gmt":"2024-10-25T15:47:21","slug":"geometry-fold-problems","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=962","title":{"rendered":"Geometry Fold Problems"},"content":{"rendered":"\n

A unit equilateral $\\triangle{ABC}$ is folded over line $DE$, forming a quadrilateral $BCDE$, with $A$ touching $BC$ at $A’$, and $\\triangle{BA’E}$ is a right triangle. The area of $BCDE$ is __________. Answer: $\\dfrac{17\\sqrt{3}-27}{8}\\approx 0.305608$<\/p>\n\n\n\n

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A quarter-circle $ABC$, with its center at $A$ and its radius as $1$, is folded over $BD$, with its center touching the arc $BC$ at $A’$. The area of the resulting figure is __________. Answer: $\\dfrac{3\\pi-2\\sqrt{3}}{12}\\approx 0.49672$<\/p>\n\n\n\n

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Two opposite vertices $A$ and $C$ of a unit square $ABCD$ are folded toward the diagonal $BD$, forming a kite-shaped area $BEDF$. The area of the kite is __________.<\/p>\n\n\n\n

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Point $A$ of a unit square $ABCD$ is folded toward the diagonal $BD$ at point $A’$. Then, point $D$ is folded over $AC$, touching point $B$, resulting in a quadrilateral $BCFE$. The area of $BCFE$ is __________.<\/p>\n\n\n\n

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Point $A$ is the center of a quarter circle $ABC$ with radius as $1$. Point $D$ is the midpoint of $AB$. Point $C$ is folded over $EF$, touching point $D$, The length of $EF$ is __________. Answer: $\\dfrac{2\\sqrt{71}-3}{40}\\sqrt{5} \\approx 0.774367$<\/p>\n\n\n\n

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Point $A$ of unit square $ABCD$ is folded over $FG$, touching point $E$, which is the midpoint of $CD$. The length of $FG$ is __________. Answer: $\\dfrac{\\sqrt{5}}{2}$<\/p>\n\n\n\n

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A semi-circle with $AB$ as its diameter, point $O$ as its center, and its radius as $1$. The circle is folded over chord $BC$ and intersecting the diameter at $O$. The area bounded by $\\overparen{AC}$, $AB$ and $BC$ is __________. Answer: $\\dfrac{\\pi}{6}+\\dfrac{\\sqrt{3}}{4}$<\/p>\n\n\n\n

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A semi-circle with $AB$ as its diameter, point $O$ as its center, and its radius as $5$. The circle is folded over chord $BD$ and intersecting the diameter at $C$, and $\\dfrac{AC}{BC}=\\dfrac{1}{2}$. The length of $BD$ is __________ (PUMaC 2010).<\/p>\n\n\n\n

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A semi-circle with $AB$ as its diameter, point $O$ as its center, and its radius as $2$. The circle is folded over chord $CD$, tangent with the $AB$ at $E$, with $\\dfrac{BE}{AE}=\\dfrac{1}{3}$. The length of $CD$ is __________.<\/p>\n\n\n\n

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$\\triangle{ABC}$ is a right triangle with $AB=BC=12$, and $\\angle{ABC}=90^\\circ$. $D$ is the midpoint of $BC$. Point $A$ is folded over $EF$ to touch point $D$. The length of $EF$ is __________.<\/p>\n\n\n\n

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$\\triangle{ABC}$ is a right triangle with $AB=3$, $BC=4$, and $\\angle{ABC}=90^\\circ$. $D$ is the midpoint of $BC$. Point $A$ is folded over $EF$ to touch point $D$. The length of $EF$ in the simplest form is $\\dfrac{a}{b}\\sqrt{c}$, where $a$, $b$, and $c$ are integers. The value of $a+b+c$ is __________.<\/p>\n\n\n\n

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$\\triangle{ABC}$ is a right triangle with $AB=BC=8$, and $\\angle{ABC}=90^\\circ$. $D$ is a point on $AC$ so that $\\dfrac{CD}{AC}=\\dfrac{1}{3}$. Point $B$ is folded over $EF$ to touch point $D$, resulting a concave polygon $A’EFCD$. The length of $EF$ is __________.<\/p>\n\n\n\n

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A pentagon was folded from a square of paper, as shown in the figure. At first the edges $BC$ and $DC$ were folded to the diagonal $AC$, so that the corners $B$ and $D$ lie on the diagonal and then the resulting shape was folded so that the vertex $C$ coincided with the vertex A. The value of the angle indicated by the question mark is __________.<\/p>\n\n\n\n

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A paper strip is folded three times as shown. If $\\alpha=70^\\circ$, then $\\beta=$ __________ .<\/p>\n\n\n\n

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In a unit equalaterial $\\triangle{ABC}$, point $A$ is folded to the point $D$ on $BC$ as shown, resulting in the crease $EF$ with $E$ on $AB$ and $F$ on $AC$. If $FD\\perp BC$: