PAS MATEMATIKA BLOG: Contoh Soal Bangun Ruang Dimensi Tiga (Matematika Wajib Kelas XII)

$\begin{array}{ll}\\ 1.&\textrm{Diketahui sebuah kubus ABCD.EFGH dengan rusuk}\: \: 8\sqrt{2}\: \: \textrm{cm}\\ &\textrm{Perhatkanlah pernyataan-pernyataan berikut berkaitan kubus di atas}!\\ &(\textrm{i})\quad \textrm{Jarak titik A ke titik C adalah 16 cm}\\ &(\textrm{ii})\: \: \: \textrm{Jarak titik A ke titik G adalah 16 cm}\\ &(\textrm{iii})\: \: \textrm{Jarak titik A ke garis CD adalah}\: \: 8\sqrt{2}\: \: \textrm{cm}\\ &(\textrm{iv})\: \: \textrm{Jarak titik A ke garis BC adalah}\: \: 8\sqrt{3}\: \: \textrm{cm}\\ &\textrm{Pernyataan yang tepat dari pernyataan-pernyataan di atas ditunjukkan oleh}....\\ &\begin{array}{lllllll}\\ \textrm{a}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{ii})&&&\textrm{d}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \textrm{b}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{iii})&\textrm{c}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iii})&\textrm{e}.&(\textrm{iii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{b}\\ &\textrm{Perhatikanlah ilustrasi gambar berikut}! \end{array}$

$.\qquad \: \begin{array}{|c|c|l|c|c|}\hline \textrm{No}&\textrm{Pilihan}&\textrm{Keterangan}&\textrm{Opsi}&\textrm{Simpulan}\\\hline 1&(\textrm{i})&\begin{aligned}AC&=a\sqrt{2}\\ &=\left ( 8\sqrt{2} \right )\sqrt{2}=16\: \: \textrm{cm} \end{aligned}&16\: \: \textrm{cm}&\textrm{Benar}\\\hline 2&(\textrm{ii})&\begin{aligned}AG&=a\sqrt{3}\\ &=\left ( 8\sqrt{2} \right )\sqrt{3}=8\sqrt{6}\: \: \textrm{cm} \end{aligned}&16\: \: \textrm{cm}&\textrm{Salah}\\\hline 3&(\textrm{iii})&\begin{aligned}A&\: \: \textrm{ke garis}\: \: CD\: \: \textrm{sama dengan}\\ A&\: \: \textrm{ke}\: \: D,\: \textrm{sama dengan sisi kubus}\\ \textrm{y}&\textrm{aitu}\: \: AD=a=8\sqrt{2}\: \: \textrm{cm} \end{aligned}&8\sqrt{2}\: \: \textrm{cm}&\textrm{Benar}\\\hline 4&(\textrm{iv})&\begin{aligned}A&\: \: \textrm{ke garis}\: \: BG\: \: \textrm{sama dengan}\\ A&\: \: \textrm{ke}\: \: B,\: \textrm{sama dengan sisi kubus}\\ \textrm{y}&\textrm{aitu}\: \: AB=a=8\sqrt{2}\: \: \textrm{cm} \end{aligned}&8\sqrt{3}\: \: \textrm{cm}&\textrm{Salah}\\\hline \end{array}$

$\begin{array}{ll}\\ 2.&\textrm{Misalkan diketahui sebuah balok ABCD.EFGH}\\ &\textrm{sebagaimana ilustrasi berikut} \end{array}$

$.\quad \, \begin{array}{ll}\\ &(\textrm{i})\quad \textrm{Jarak titik A ke titik B adalah 16 cm}\\ &(\textrm{ii})\: \: \: \textrm{Jarak titik A ke titik C adalah}\: \: 4\sqrt{41}\: \: \textrm{cm}\\ &(\textrm{iii})\: \: \textrm{Jarak titik A ke garis CD adalah}\: \: 16\: \: \textrm{cm}\\ &(\textrm{iv})\: \: \textrm{Jarak titik A ke bidang BFGC adalah}\: \: 12\: \: \textrm{cm}\\ &\\ &\textrm{Pernyataan di atas yang tepat ditunjukkan oleh}....\\ &\begin{array}{lllllll}\\ \textrm{a}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{ii})&&&\textrm{d}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \textrm{b}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{iii})&\textrm{c}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iii})&\textrm{e}.&(\textrm{iii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{c}\\ & \end{array}$

$.\qquad \: \, \begin{array}{|c|c|l|c|c|}\hline \textrm{No}&\textrm{Pilihan}&\qquad\qquad\textrm{Keterangan}&\textrm{Opsi}&\textrm{Keterangan}\\\hline 1&(\textrm{i})&\begin{aligned}AB&=20\: \: \textrm{cm} \end{aligned}&16\: \: \textrm{cm}&\textrm{Salah}\\\hline 2&(\textrm{ii})&\begin{aligned}AC&=\sqrt{AB^{2}+BC^{2}}\\ &=\sqrt{20^{2}+16^{2}}\\ &=\sqrt{400+256}=\sqrt{656}\\ &=4\sqrt{41}\: \: \textrm{cm} \end{aligned}&4\sqrt{41}\: \: \textrm{cm}&\color{blue}\textrm{Benar}\\\hline 3&(\textrm{iii})&\begin{aligned}A&\: \: \textrm{ke garis}\: \: CD=\: A\: \textrm{ke}\: \: D\\ =&AD=16\: \: \textrm{cm} \end{aligned}&16\: \: \textrm{cm}&\color{blue}\textrm{Benar}\\\hline 4&(\textrm{iv})&\begin{aligned}A&\: \: \textrm{ke bidang}\: \: BCGF\\ =&\: \textrm{jarak}\: \: A\: \: \textrm{ke}\: \: B\\ =&AB=20\: \: \textrm{cm} \end{aligned}&12\: \: \textrm{cm}&\textrm{Salah}\\\hline \end{array}$

$\begin{array}{ll}\\ 3.&\textrm{Jarak bidang ACH dengan EGB pada kubus ABCD.EFGH}\\ &\begin{array}{lllllll}\\ \textrm{a}.&4\sqrt{3}&&&\textrm{d}.&6\\ \textrm{b}.&2\sqrt{3}\qquad&\textrm{c}.&4\qquad&\textrm{e}.&12\\ \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{d}\\ &\begin{aligned}\textrm{Luas}&\: \textrm{jajar genjang BQHP = luas jajar genjang BQHP}\\ \textrm{alas}&\times \textrm{tinggi}=\textrm{alas}\times \color{blue}tinggi\\ \textrm{HP}&\times \textrm{PQ}=\textrm{HQ}\times \color{blue}tinggi\: \: (\textbf{jarak garis HQ ke garis PB})\\ \textrm{Jarak}&\times \textrm{HQ}=\textrm{HP}\times \textrm{PQ}\\ \textrm{Jarak}&=\displaystyle \frac{HP\times PQ}{HQ}\\ &=\displaystyle \frac{\displaystyle \frac{1}{2}a\sqrt{2}\times a}{\displaystyle \frac{1}{2}a\sqrt{6}}=\displaystyle \frac{a\sqrt{2}}{\sqrt{6}},\qquad \textrm{diketahui}\: \: a=\textrm{sisi}=6\sqrt{3}\: \: \textrm{cm}\\ &=\displaystyle \frac{\left ( 6\sqrt{3}\: \: \textrm{cm} \right )\sqrt{2}}{\sqrt{6}}=6\: \: \textrm{cm} \end{aligned} \end{array}$

Sebagai tambahan penjelasan berikut diilustrasikan panjang beberapa unsur dalam kubus

$\begin{array}{ll}\\ 4.&\textrm{Diketahui kubus ABCD.EFGH dengan rusuk}\: \: 3\sqrt{2}\: \: \textrm{cm}\\ &\textrm{dan diberikan pernyataan-pernyataan berikut}\\ &(\textrm{i})\quad \textrm{Jarak titik A ke titik G adalah 6 cm}\\ &(\textrm{ii})\: \: \: \textrm{Jarak titik A ke garis CD adalah}\: \: 3\sqrt{2}\: \: \textrm{cm}\\ &(\textrm{iii})\: \: \textrm{Jarak titik A ke garis CG adalah 6 cm}\\ &(\textrm{iv})\: \: \textrm{Jarak titik A ke titik E adalah 6 cm}\\\\ &\textrm{Pernyataan yang tepat dari pernyataan-pernyataan}\\ & \textrm{di atas ditunjukkan oleh}....\\ &\begin{array}{lllllll}\\ \textrm{a}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{ii})&&&\textrm{d}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \textrm{b}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{iii})&\textrm{c}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iii})&\textrm{e}.&(\textrm{iii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{c}\\ & \end{array}$

$.\qquad \begin{array}{|c|c|l|c|c|}\hline \textrm{No}&\textrm{Pilihan}&\textrm{Keterangan}&\textrm{Opsi}&\textrm{Simpulan}\\\hline 1&(\textrm{i})&\begin{aligned}AG&=a\sqrt{3}=\left ( 3\sqrt{2} \right )\sqrt{3}\\ &=3\sqrt{6}\: \: \textrm{cm} \end{aligned}&6\: \: \textrm{cm}&\textrm{Salah}\\\hline 2&(\textrm{ii})&\begin{aligned}A&\: \: \textrm{ke garis}\: \: CD=\overline{AD}.\\ &=3\sqrt{2}\: \: \textrm{cm}\\ \end{aligned}&3\sqrt{2}\: \: \textrm{cm}&\color{blue}\textrm{Benar}\\\hline 3&(\textrm{iii})&\begin{aligned}A&\: \: \textrm{ke garis}\: \: CG=\overline{AC}.\\ &\overline{AC}=a\sqrt{2}\\ &\overline{AC}=\left ( 3\sqrt{2} \right )\sqrt{2}=6 \end{aligned}&6\: \: \textrm{cm}&\color{blue}\textrm{Benar}\\\hline 4&(\textrm{iv})&\begin{aligned}A&\: \: \textrm{ke titik}\: \: E=\overline{AE}=3\sqrt{2} \end{aligned}&6\: \: \textrm{cm}&\textrm{Salah}\\\hline \end{array}$

$\begin{array}{ll}\\ 5.&\textrm{Diketahui kubus ABCD.EFGH dengan rusuk}\: \: a\: \: \textrm{cm}\\ &\textrm{Jarak titik A ke diagonal HB adalah}....\\ &\begin{array}{lllllll}\\ \textrm{a}.&\displaystyle \frac{a}{3}\sqrt{2}&&&\textrm{d}.&\displaystyle \frac{a}{3}\sqrt{6}\\\\ \textrm{b}.&\displaystyle \frac{a}{3}\sqrt{3}\qquad&\textrm{c}.&\displaystyle \frac{a}{3}\sqrt{5}\qquad&\textrm{e}.&\displaystyle \frac{a}{3}\sqrt{7}\\ \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{d}\\ & \end{array}$

$.\quad\: \color{blue}\textrm{Perhatikanlah ilustrasi berikut ini}$

dan

$.\qquad\begin{aligned}\textbf{Luas}\Delta \textrm{AHB}&=\textbf{luas}\Delta \textrm{AHB}\\ \displaystyle \frac{1}{2}\times \textrm{alas}\times \color{blue}\textrm{tinggi}&=\displaystyle \frac{1}{2}\times \textrm{alas}\times \textrm{tinggi}\\ \textrm{HB}\times \color{blue}\textrm{t}&=\textrm{AB}\times \textrm{AH}\\ \color{blue}\textrm{t}&=\displaystyle \frac{\textrm{AB}\times \textrm{AH}}{\textrm{HB}}\\ &=\displaystyle \frac{a\times a\sqrt{2}}{a\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}\\ &=\displaystyle \frac{a\sqrt{6}}{3}\\ &=\displaystyle \frac{a}{3}\sqrt{6} \end{aligned}$

PAS MATEMATIKA BLOG

Contoh Soal Bangun Ruang Dimensi Tiga (Matematika Wajib Kelas XII)

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