Exercise Zone : Balok (Cuboid)

Berikut ini adalah kumpulan soal mengenai Balok Tingkat Dasar. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

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No. 1

Balok $ABCD.EFGH$ alasnya persegi dengan ${AB = 4}$ cm, ${AE = 8}$ cm. Jika $\theta$ adalah sudut antara bidang $ACH$ dan bidang $ABCD$, maka $\sin\theta=$

1. $\dfrac45\sqrt2$
2. $\dfrac23\sqrt2$
3. $\dfrac12\sqrt2$

4. $\dfrac47\sqrt2$
5. $\dfrac13$

Ganesha Operation

$DB=4\sqrt2$

$\begin{array}{rl}DO& ={\displaystyle \frac{1}{2}}DB\\ & ={\displaystyle \frac{1}{2}}\left(4\sqrt{2}\right)\\ & =2\sqrt{2}\end{array}$

$\begin{array}{rl}HO& =\sqrt{H{D}^2+D{O}^2}\\ & =\sqrt{8^2+{\left(2\sqrt{2}\right)}^2}\\ & =\sqrt{64+8}\\ & =\sqrt{72}\\ & =6\sqrt{2}\end{array}$

${\theta=\angle(ACH,ABCD)=\angle(HO,DO)}$

$\begin{array}{rl}\sin \theta & ={\displaystyle \frac{HD}{HO}}\\ & ={\displaystyle \frac{8}{6\sqrt{2}}}\times {\displaystyle \frac{\sqrt{2}}{\sqrt{2}}}\\ & ={\displaystyle \frac{8}{12}}\sqrt{2}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle {\displaystyle \frac{2}{3}}\sqrt{2}}}}}\end{array}$

Jadi, $\sin\theta=\dfrac23\sqrt2$.
JAWAB: B

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No. 2

Balok $PQRS.TUVW$ mempunyai panjang $PQ$ = 6 cm. $PS$ = 4 cm, dan $PT$ = 3 cm. Titik $X$ merupakan perpotongan antara diagonal $QV$ dan $RU$. Jarak antara titik $P$ dan titik $X$ adalah ....

1. $7{,}5$ cm
2. $7{,}25$ cm
3. $7{,}0$ cm

4. $6{,}5$ cm
5. $6{,}25$ cm

Bot tanyamtk

$\begin{array}{rl}QV& =\sqrt{Q{R}^2+R{V}^2}\\ & =\sqrt{4^2+3^2}\\ & =\sqrt{16+9}\\ & =\sqrt{25}\\ & =5\end{array}$

$\begin{array}{rl}QX& ={\displaystyle \frac{1}{2}}QV\\ & ={\displaystyle \frac{1}{2}}(5)\\ & ={\displaystyle \frac{5}{2}}\end{array}$

$\begin{array}{rl}PX& =\sqrt{P{Q}^2+Q{X}^2}\\ & =\sqrt{6^2+{\left({\displaystyle \frac{5}{2}}\right)}^2}\\ & =\sqrt{36+{\displaystyle \frac{25}{4}}}\\ & =\sqrt{{\displaystyle \frac{144}{4}}+{\displaystyle \frac{25}{4}}}\\ & =\sqrt{{\displaystyle \frac{169}{4}}}\\ & ={\displaystyle \frac{13}{2}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 6,5}}}}\end{array}$

Jadi, jarak antara titik $P$ dan titik $X$ adalah $6{,}5$ cm.
JAWAB: D

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