Diketahui segitiga ABC di mana \( A(2x, 7, 3), B(x, 7, 7) \) dan \( C(10,16,3x) \). Jika segitiga ABC siku-siku di A maka nilai \(x= \cdots\)
- \( -5 \)
- \( -4 \)
- \( 1 \)
- \( 2 \)
- \( 4 \)
Pembahasan:
Pertama, kita cari dulu vektor \( \overrightarrow{AB} \) dan \( \overrightarrow{AC} \), yakni:
\begin{aligned} \overrightarrow{AB} &= B - A = (x, 7, 7) - (2x,7,3) \\[8pt] &= (x-2x, \ 7-7, \ 7-3) = (-x, \ 0, \ 4) \\[8pt] \overrightarrow{AC} &= C - A = (10,16,3x) - (2x,7,3) \\[8pt] &= (10-2x, \ 16-7, \ 3x-3) = (10-2x, \ 9, \ 3x-3) \end{aligned}
Karena segitiga ABC siku-siku di A maka sudut \( \overrightarrow{AB} \) dan \( \overrightarrow{AC} \) adalah \( 90^\circ \) sehingga berlaku:
\begin{aligned} \overrightarrow{AB} \cdot \overrightarrow{AC} &= 0 \\[8pt] (-x, \ 0, \ 4) \cdot (10-2x, \ 9, \ 3x-3) &= 0 \\[8pt] (-x)(10-2x)+(0)(9)+(4)(3x-3) &= 0 \\[8pt] -10x+2x^2+0+12x-12 &= 0 \\[8pt] 2x^2+2x-12 &= 0 \\[8pt] x^2+x-6 &= 0 \\[8pt] (x+3)(x-2) &= 0 \\[8pt] x = -3 \ \text{atau} \ x &= 2 \end{aligned}
Jawaban D.