Fun Geometry Problem with Solution #50



٨ x = 6°
٨(¤س Freezer Freezer)



(1)ACB = 36°РADB = 66°

(2) ˹ش P AB DP ⊥ AB => APD = 90°РBPD = 90° <=> ADP = 6°РBDP = 60°
ҠBDP 繠∆ҡ ՠP ҡ РB = 30° => BD = 2・DP

(3) ˹ش Q ͡ABD DQ = DP РADQ = 6°
ҠADQ ADP ¤ѹẺ -- (DQ = DP,ADQ =ADP, AD = AD) => AQD =APD <=> AQD = 90°

(4) CA ͡仾ǹ͢¢ͧ DQ ش R => ARD = 6° <=> ARD =ADR <=> ADR 繠˹Ҩ ՠA ʹ
AQD = 90° <=> AQ ǹ٧ͧADR <=> DQ = QR
DR = 2DQ <=> DR = 2DP <=> DR = BD

(5) ѧࡵ BD = DR РBDR = 2(BCR) => ش D 繨شٹҧͧǧՠBCR Ṻ => CD = DR <=> CDR 繠˹Ҩ ՠD ʹ <=> DCR =CRD <=> DCR = 6° Q.E.D.



Create Date : 19 ѹ¹ 2557
Last Update : 19 ѹ¹ 2557 0:01:59 .
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Fun Geometry Problem with Solution #49
blog йʹ⨷ Ҩ繺 (Lemma) Ѻþ٨駵




٨ AC = BD = CD
٨



(1) BA ͡ѧش P ·∠BPC= α <=> ∠BPC=∠CBP <=> ∆BCP繠∆˹Ҩ ՠ∠Cʹ <=> CP = BC
͡ҡ ѧҠ∠ACP= α <=> ∠ACP=∠APC <=> ∆ACP繠∆˹Ҩ ՠ∠Aʹ <=> AP = AC <=> AP = BD

(2) ѧࡵҠ∆BCD ∆ACP¤ѹẺ -- (BD = AP,∠CBD=∠APC, BC = CP) => CD= AC
∴ AC = BD = CD Q.E.D.



Create Date : 16 ѹ¹ 2557
Last Update : 16 ѹ¹ 2557 0:00:31 .
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Fun Geometry Problem with Solution #48



٨ x = 6°
٨



(1) BC ͡ѧش P ·BPD = 90° <=> BDP = 60° <=> CDP = 60°- 3x
ҠBDP 繠∆ҡ ՠP ҡ РB = 30° => BD = 2・DP <=> AD = 2・DP

(2)AC = CD <=> ACD 繠˹Ҩ ՠC ʹ

(3) ˹ش Q AD CQD = 90° <=> CQ ǹ٧ͧACD <=> DQ = (AD)/2 = DP

(4) ѧࡵҠCDP CDQ ¤ѹẺ -- (CPD =CQD = 90°, DP = DQ, CD = CD) => CDP =CDQ <=> 60°- 3x = 7x <=> x = 6° Q.E.D.



Create Date : 13 ѹ¹ 2557
Last Update : 13 ѹ¹ 2557 0:00:02 .
Counter : 652 Pageviews.

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Fun Geometry Problem with Solution #47



٨ x = 6°
٨( س Didy Ricra Osorio)



(1)BDC = 3x

(2) ˹ش P AC DP = AD <=> ADP 繠˹Ҩ ՠD ʹ <=> APD =DAP <=> APD = 2x <=> CDP = x

(3) ˹ش Q AB DAQ = x РAQD = x <=> ADQ 繠˹Ҩ ՠD ʹ <=> AD = DQ
͡ҡ ѧҠBDQ = 2x
ѧࡵҠADQ CDP ¤ѹẺ -- (AQD =DCP,DAQ =CDP, AD = DP) => AQ = CD
ѧࡵҠBDQ ACD ¤ѹẺ -- (BD = AC,BDQ =CAD, DQ = AD) => BQ = CD РDBQ (=ACD) = x

(4) ѧࡵҠACQ BCD ¤ѹẺ -- (AC = BD,CAQ =BDC, AQ = CD) => ACQ (=CBD) = 5x <=> DCQ = 4x
͡ҡ ѧ CQ = BC <=> BCQ 繠˹Ҩ ՠC ʹ <=> BQC =CBQ <=> BQC = 6x

(5) ˹ش R CD DCR = 6x РCDR = 6x <=> QCR = 2x РQDR = x
ѧࡵҠCDR BCQ ¤ѹẺ -- (DCR =CBQ, CD = BQ,CDR=BQC) => CR = BC DR = CQ (= BC) => CQ = CR = DR
ѧࡵҠ☐CQDR CQ = CR = DR,∠C = 2x Р∠D = x => ∠CRD = 120°- 2x (ԡʹԸաþ٨⨷ 1)

(6) ԨóҠCDR ҠDCR +CDR +CRD = 180° <=> 6x + 6x + (120°- 2x) = 180° <=> x = 6° Q.E.D.



Create Date : 10 ѹ¹ 2557
Last Update : 10 ѹ¹ 2557 0:00:00 .
Counter : 818 Pageviews.

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Fun Geometry Problem with Solution #46



٨ x = 15°
٨



(1)ACD = 90°- 5x РBDC = 90°- 3x

(2) ˹ش P 㹠BCD DBP = 2x РBDP = 90°- 5x <=> CBP = x РCDP = 2x
ҠBDP ACD ¤ѹẺ -- (DBP =CAD, BD = AC,BDP =ACD) => DP = CD <=> CDP 繠˹Ҩ ՠD (= 2x) ʹ <=> DCP (=CPD) = 90°- x <=> BCP = x <=> BCP =CBP <=> BCP 繠˹Ҩ ՠP ʹ <=> BP = CP
͡ҡ ѧ BP = AD <=> CP = AD

(3) AB ͡ѧش Q ·AQC = 2x <=> AQC =CAQ <=> ACQ 繠˹Ҩ ՠC ʹ <=> AC = CQ
͡ҡ ѧҠBCQ = x

(4) ˹ش R CQ CR = AD
ѧࡵҠBCR BCP ¤ѹẺ -- (CR = CP,BCR =BCP, BC = BC) => CBR (=CBP) = x <=> BRQ = 2x <=> BRQ =BQR <=> BQR 繠˹Ҩ ՠB ʹ <=> BQ = BR
͡ҡ ѧ BR = BP <=> BQ = AD

(5) ѧࡵҠACD BCQ ¤ѹẺ -- (AC = CQ,CAD =BQC, AD = BQ) => ACD =BCQ <=> 90°- 5x = x <=> x = 15° Q.E.D.



Create Date : 07 ѹ¹ 2557
Last Update : 7 ѹ¹ 2557 0:00:02 .
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