`(0, 0, 0), (1,2, 3)` और `(-3, -2, 1)` शीर्षो वाले त्रिभुज का क्षेत्रफल क्या है ?

Options

A. `3sqrt(5)` वर्ग इकाई
B. `6sqrt(5)` वर्ग इकाई
C. 6 वर्ग इकाई
D. 12 वर्ग इकाई

Answer

A

Explanation

माना `triangleABC` के शीर्ष निम्न है |
`A to (x_(1), y_(1), z_(1)) to (0, 0, 0)`
`B to (x_(2), y_(2), y_(2)) to (1, 2, 3)`
`C to (x_(3), y_(3), z_(3)) to (-3, -2, 1)`
अब, `triangle_(z)=(1)/(2)|{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}|=(1)/(2)|{:(0,,0,,1),(1,,2,,1),("-3",,"-2",,1):}|`
`= (1)/(2) (-2 + 6) = (4)/(2) =2`
`triangle_(y)=(1)/(2)|{:(z_(1),,x_(1),,1),(z_(2),,x_(2),,1),(z_(3),,x_(3),,1):}|=(1)/(2)|{:(0,,0,,1),(3,,1,,1),("1",,"-3",,1):}|`
` = (1)/(2)(-9 -1) = (1)/(2) xx (-10) = -5`
`triangle_(x)=(1)/(2)|{:(y_(1),,z_(1),,1),(y_(2),,z_(2),,1),(y_(3),,z_(3),,1):}|=(1)/(2)|{:(0,,0,,1),(2,,3,,1),("-2",,"1",,1):}|`
`= (1)/(2) (2 + 6) = (8)/(2) = 4`
`therefore triangleABC` का अभीष्ट क्षेत्रफल
`=sqrt(triangle_(x)^(2)+triangle_(y)^(2)+triangle_(z)^(2))=sqrt((2)^(2)+(-5)^(2)+(4)^(2))`
`=sqrt(4 + 25 + 16) = 3sqrt(5)` वर्ग इकाई