:HL["https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image14-MuiHG6V15lFf1IyZYqFsT5fp2ylxml.png","image"] :HL["https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image16-cAwoDcOx9blfPpbkXYNKzTLRfm2bPL.png","image"] :HL["https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image19-1gfm65SmhktTYj7uGyaEzzWTkoqZCH.png","image"] :HL["https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image22-3iPw4ppbshnL9RkGZ2SYp7hrnBz3ky.png","image"] 2a:["$","span","p0-0-t0-m3",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\left( 1 , 2 \\right)

"}}] 2b:["$","span","p0-0-t0-m5",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

y

"}}] 2c:["$","span","p0-0-t0-m7",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

4

"}}] 8a:T506,

\displaystyle \frac{9}{2}

2d:["$","span","p0-0-t0-m9",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$8a"}}] 2e:["$","span","p0-0-t0-m11",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

5

"}}] 8b:T509,

\displaystyle \frac{11}{2}

2f:["$","span","p0-0-t0-m13",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$8b"}}] 30:["$","span","p0-0-t0-m15",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

6

"}}] 8c:T1495,

\left( {\vec{\mathrm{OP}}} - {\vec{\mathrm{OA}}} \right) \bullet {\vec{\mathrm{OB}}} = 0

31:["$","span","p0-0-t0-m9",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$8c"}}] 8d:Tc92,

\left| {\vec{\mathrm{OP}}} \right|

32:["$","span","p0-0-t0-m11",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$8d"}}] 8e:T6b9,

\displaystyle \sqrt{2}

33:["$","span","p0-0-t0-m13",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$8e"}}] 8f:T6b9,

\displaystyle \sqrt{3}

34:["$","span","p0-0-t0-m15",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$8f"}}] 35:["$","span","p0-0-t0-m17",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

2

"}}] 90:T6b9,

\displaystyle \sqrt{5}

36:["$","span","p0-0-t0-m19",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$90"}}] 91:T6b9,

\displaystyle \sqrt{6}

37:["$","span","p0-0-t0-m21",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$91"}}] 92:T5ca,

{\mathrm{F}} \left( c , 0 \right) \left( c > 0 \right)

39:["$","span","p0-0-t0-m3",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$92"}}] 3a:["$","span","p0-0-t0-m5",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

y

"}}] 3b:["$","span","p0-0-t0-m7",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

1

"}}] 3c:["$","span","p0-0-t0-m9",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{P}}

"}}] 93:T54b,

\displaystyle {\overline{\mathrm{PF}}} = 5

3d:["$","span","p0-0-t0-m11",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$93"}}] 3e:["$","span","p0-0-t0-m13",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

b ^{2}

"}}] 3f:["$","span","p0-0-t0-m15",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

b

"}}] 40:["$","span","p0-0-t0-m17",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

16

"}}] 41:["$","span","p0-0-t0-m19",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

18

"}}] 42:["$","span","p0-0-t0-m21",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

20

"}}] 43:["$","span","p0-0-t0-m23",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

22

"}}] 44:["$","span","p0-0-t0-m25",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

24

"}}] 45:["$","span","p0-0-t0-m15",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{A}}

"}}] 46:["$","span","p0-0-t0-m17",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{B}}

"}}] 47:["$","span","p0-0-t0-m19",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

l

"}}] 48:["$","span","p0-0-t0-m21",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{C}}

"}}] 49:["$","span","p0-0-t0-m23",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{A}}

"}}] 4a:["$","span","p0-0-t0-m25",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{B}}

"}}] 4b:["$","span","p0-0-t0-m27",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{C}}

"}}] 4c:["$","span","p0-0-t0-m29",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{P}}

"}}] 94:T54b,

\displaystyle {\overline{\mathrm{AP}}} = 3

4d:["$","span","p0-0-t0-m31",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$94"}}] 95:Ta66,

\displaystyle {\overline{\mathrm{BP}}} = 3 \sqrt{2}

4e:["$","span","p0-0-t0-m33",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$95"}}] 96:T1c3d,

\displaystyle \frac{{\overline{\mathrm{AP}}}}{{\overline{\mathrm{CA}}}} = \frac{{\overline{\mathrm{BP}}}}{{\overline{\mathrm{BA}}}} = \frac{{\overline{\mathrm{BA}}}}{{\overline{\mathrm{BC}}}}

4f:["$","span","p0-0-t0-m35",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$96"}}] 50:["$","span","p0-0-t0-m37",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{P}}

"}}] 51:["$","span","p0-0-t0-m39",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{BC}}

"}}] 52:["$","span","p0-0-t0-m41",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{H}}

"}}] 53:["$","span","p0-0-t0-m43",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{PH}}

"}}] 97:T6e0,

\displaystyle 2 \sqrt{2}

54:["$","span","p0-0-t0-m45",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$97"}}] 98:T6bc,

\displaystyle \sqrt{10}

55:["$","span","p0-0-t0-m47",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$98"}}] 99:T6e0,

\displaystyle 2 \sqrt{3}

56:["$","span","p0-0-t0-m49",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$99"}}] 9a:T6bc,

\displaystyle \sqrt{14}

57:["$","span","p0-0-t0-m51",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$9a"}}] 58:["$","span","p0-0-t0-m53",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

4

"}}] 59:["$","p","p0-1",{"className":"mt-0 mb-[1.2em] [tab-size:3] whitespace-pre-wrap","children":[["$","img","p0-1-i2",{"src":"https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image14-MuiHG6V15lFf1IyZYqFsT5fp2ylxml.png","alt":"","className":"problem-img my-[1em] block","style":{"--img-w":"83cqw"}}]]}] 9b:T405,

C _{1}

5b:["$","span","p0-0-t0-m7",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$9b"}}] 5c:["$","span","p0-0-t0-m9",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

1

"}}] 5d:["$","span","p0-0-t0-m11",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{P}}

"}}] 5e:["$","span","p0-0-t0-m13",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

x

"}}] 9c:T405,

C _{2}

5f:["$","span","p0-0-t0-m15",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$9c"}}] 9d:T405,

C _{1}

60:["$","span","p0-0-t0-m17",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$9d"}}] 9e:T405,

C _{2}

61:["$","span","p0-0-t0-m19",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$9e"}}] 62:["$","span","p0-0-t0-m21",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

x

"}}] 63:["$","span","p0-0-t0-m23",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{Q}}

"}}] 64:["$","span","p0-0-t0-m25",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{Q}}

"}}] 9f:T405,

C _{2}

65:["$","span","p0-0-t0-m27",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$9f"}}] 66:["$","span","p0-0-t0-m29",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{H}}

"}}] a0:Ta69,

\displaystyle {\overline{\mathrm{PH}}} = 4 \sqrt{15}

67:["$","span","p0-0-t0-m31",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a0"}}] a1:Tb06,

\displaystyle {\overline{\mathrm{QH}}} = 5 \sqrt{6}

68:["$","span","p0-0-t0-m33",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a1"}}] 69:["$","span","p0-0-t0-m35",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{PF}}

"}}] 6a:["$","span","p0-0-t0-m37",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{P}}

"}}] 6b:["$","span","p0-0-t0-m39",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

x

"}}] 6c:["$","span","p0-0-t0-m41",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{F}}

"}}] 6d:["$","span","p0-0-t0-m43",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

x

"}}] a2:T50f,

\displaystyle \frac{389}{40}

6e:["$","span","p0-0-t0-m45",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a2"}}] a3:T50f,

\displaystyle \frac{197}{20}

6f:["$","span","p0-0-t0-m47",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a3"}}] a4:T50f,

\displaystyle \frac{399}{40}

70:["$","span","p0-0-t0-m49",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a4"}}] a5:T50f,

\displaystyle \frac{101}{10}

71:["$","span","p0-0-t0-m51",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a5"}}] a6:T50f,

\displaystyle \frac{409}{40}

72:["$","span","p0-0-t0-m53",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a6"}}] 73:["$","p","p0-1",{"className":"mt-0 mb-[1.2em] [tab-size:3] whitespace-pre-wrap","children":[["$","img","p0-1-i2",{"src":"https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image16-cAwoDcOx9blfPpbkXYNKzTLRfm2bPL.png","alt":"","className":"problem-img my-[1em] block","style":{"--img-w":"79cqw"}}]]}] 74:["$","span","p0-0-t0-m15",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{D}}

"}}] a7:T7b3,

\displaystyle {\overline{\mathrm{BC}}} = {\overline{\mathrm{CD}}}

75:["$","span","p0-0-t0-m17",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a7"}}] a8:T54e,

\displaystyle {\overline{\mathrm{BD}}} = 10

76:["$","span","p0-0-t0-m19",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a8"}}] a9:Ta42,

\displaystyle {\overline{\mathrm{AC}}} = \sqrt{74}

77:["$","span","p0-0-t0-m21",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$a9"}}] aa:T7d5,

\displaystyle {\overline{\mathrm{AB}}} < {\overline{\mathrm{AD}}}

78:["$","span","p0-0-t0-m23",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$aa"}}] 79:["$","span","p0-0-t0-m25",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{OA}}

"}}] 7a:["$","span","p0-0-t0-m27",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{BCD}}

"}}] 7b:["$","span","p0-0-t0-m29",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\theta

"}}] ab:T6c9,

\displaystyle \cos \theta = \frac{3}{5}

7c:["$","span","p0-0-t0-m31",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$ab"}}] 7d:["$","span","p0-0-t0-m33",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{ABD}}

"}}] 7e:["$","span","p0-0-t0-m35",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{BCD}}

"}}] 7f:["$","p","p0-1",{"className":"mt-0 mb-[1.2em] [tab-size:3] whitespace-pre-wrap","children":[["$","img","p0-1-i2",{"src":"https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image19-1gfm65SmhktTYj7uGyaEzzWTkoqZCH.png","alt":"","className":"problem-img my-[1em] block","style":{"--img-w":"57cqw"}}]]}] 83:["$","span","p0-0-t0-m7",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{ABCD}}

"}}] ac:T2dc2,

\displaystyle \left| {\vec{\mathrm{PA}}} + {\vec{\mathrm{PB}}} + {\vec{\mathrm{PC}}} + {\vec{\mathrm{PD}}} \right| = \frac{1}{2} \left| {\vec{\mathrm{BD}}} \right|

84:["$","span","p0-0-t0-m9",{"className":"my-[3px] inline-block align-middle leading-none","dangerouslySetInnerHTML":{"__html":"$ac"}}] 85:["$","span","p0-0-t0-m11",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{P}}

"}}] ad:T1345,

{\vec{\mathrm{AQ}}} = {\vec{\mathrm{AC}}} - {\vec{\mathrm{AP}}}

86:["$","span","p0-0-t0-m13",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$ad"}}] 87:["$","span","p0-0-t0-m15",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"

{\\mathrm{Q}}

"}}] ae:Td26,

{\vec{\mathrm{PB}}} \bullet {\vec{\mathrm{DQ}}}

88:["$","span","p0-0-t0-m17",{"className":"$undefined","dangerouslySetInnerHTML":{"__html":"$ae"}}] 89:["$","p","p0-1",{"className":"mt-0 mb-[1.2em] [tab-size:3] whitespace-pre-wrap","children":[["$","img","p0-1-i2",{"src":"https://2vcomivapkynmsnb.public.blob.vercel-storage.com/problems/139/image22-3iPw4ppbshnL9RkGZ2SYp7hrnBz3ky.png","alt":"","className":"problem-img my-[1em] block","style":{"--img-w":"52cqw"}}]]}]

2025년-07월-고3-수학문제(기하)