Question

xii ips - 3 soal ujian mid semester sma negeri 1 sibabangun t.p 2025 - 2026 mata pelajaran : pend. matematika umum kelas : xii. ipa - ips 1. diketahui kubus abcd.efgh dengan panjang rusuk 4 cm. titik p ditengah eh. tentukc jarak titik p ke garis bg !

Explanation:

Step1: Establish a coordinate - system

Let \(A=(0,0,0)\), \(B=(4,0,0)\), \(G=(4,4,4)\), \(E=(0,0,4)\), \(H=(0,4,4)\). Since \(P\) is the mid - point of \(EH\), then \(P=(0,2,4)\).

Step2: Write the vector equation of line \(BG\)

The vector \(\overrightarrow{BG}=(4 - 4,4 - 0,4 - 0)=(0,4,4)\), and a point on the line \(B=(4,0,0)\). The parametric equations of the line \(BG\) are \(x = 4\), \(y = 4t\), \(z = 4t\). Let \(Q=(4,4t,4t)\) be a point on the line \(BG\).

Step3: Calculate the vector \(\overrightarrow{PQ}\)

\(\overrightarrow{PQ}=(4 - 0,4t - 2,4t - 4)=(4,4t - 2,4t - 4)\).

Step4: Use the perpendicular condition

Since \(\overrightarrow{PQ}\cdot\overrightarrow{BG}=0\), we have \((4,4t - 2,4t - 4)\cdot(0,4,4)=0\).
\[4\times0+(4t - 2)\times4+(4t - 4)\times4 = 0\]
\[16t-8 + 16t-16 = 0\]
\[32t=24\]
\[t=\frac{3}{4}\]

Step5: Calculate the distance

When \(t = \frac{3}{4}\), \(Q=(4,3,3)\). Then \(\vert\overrightarrow{PQ}\vert=\sqrt{(4 - 0)^2+(3 - 2)^2+(3 - 4)^2}=\sqrt{16 + 1+1}=\sqrt{18}=3\sqrt{2}\text{ cm}\).

Answer:

\(3\sqrt{2}\text{ cm}\)