diketahui kubus abcd efgh dengan rusuk 4 cm

Diketahui kubus ABCD.EFGH dengan panjang rusuk 4 cm. Jika titik P titik tengah EH, maka gambarnya:

dari gambar di dapat lah segitiga PFC seperti berikut:

cari terlebih dahulu ukuran CF, PF, dan PC dengan cara:

– ukuran CF

Karena diagonal sisi kubus dengan rusuk r adalah r square root of 2 dan CF adalah salah satu diagonal sisi kubus dengan rusuk 4 cm, maka  C F equals 4 square root of 2 (diagonal sisi)

– ukuran PF

P F equals square root of P E squared plus E F squared end root P F equals square root of 2 squared plus 4 squared end root P F equals square root of 4 plus 16 end root P F equals square root of 20 P F equals square root of 4 cross times 5 end root P F equals 2 square root of 5

– ukuran PC

P C equals square root of P H squared plus H C squared end root P H equals square root of 2 squared plus open parentheses 4 square root of 2 close parentheses squared end root P H equals square root of 4 plus 32 end root P H equals square root of 36 P H equals 6

sehingga segitiga PFC menjadi:

dengan ukuran yang di dapat, segitiga PCF merupakan segitiga sembarang.

Misalkan QF = x, maka QC = 4 square root of 2 minus x

jarak titik P ke CF adalah PQ, dengan demikian:

– perhatikan segitiga PFQ, di dapat:

begin mathsize 14px style P Q squared equals P F squared minus F Q squared P Q squared equals open parentheses 2 square root of 5 close parentheses squared minus x squared P Q squared equals 20 minus x squared space... space left parenthesis 1 right parenthesis end style

– perhatikan segitiga PQC, di dapat:

begin mathsize 14px style P Q squared equals P C squared minus C Q squared P Q squared equals 6 squared minus open parentheses 4 square root of 2 minus x close parentheses squared P Q squared equals 36 minus 32 plus 8 square root of 2 x minus x squared P Q squared equals 4 plus 8 square root of 2 x minus x squared space... space left parenthesis 2 right parenthesis end style

Persamaan (1) sama dengan persamaan (2), maka:

begin mathsize 14px style space space space space space space space space space 20 minus x squared equals 4 plus 8 square root of 2 x minus x squared 4 plus 8 square root of 2 x minus x squared equals 20 minus x squared space space space space space space space space space space 8 square root of 2 x equals 20 minus x squared minus 4 plus x squared space space space space space space space space space space 8 square root of 2 x equals 16 space space space space space space space space space space space space space space space space x equals fraction numerator 16 over denominator 8 square root of 2 end fraction space space space space space space space space space space space space space space space space x equals fraction numerator 2 over denominator square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction space space space space space space space space space space space space space space space space x equals fraction numerator 2 square root of 2 over denominator 2 end fraction space space space space space space space space space space space space space space space space x equals square root of 2 end style

Substitusikan nilai x ke persamaan (1), di dapat:

begin mathsize 14px style P Q squared equals 20 minus x squared P Q squared equals 20 minus open parentheses square root of 2 close parentheses squared P Q squared equals 20 minus 2 P Q squared equals 18 space P Q equals square root of 18 end style

Jadi, jawaban yang benar adalah B.

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