Question:
The two vertices of a triangle are $(4,2,1)$ and $(5,1,4)$. If the centroid is $(5,2,3)$, then the third vertex is
The two vertices of a triangle are $(4,2,1)$ and $(5,1,4)$. If the centroid is $(5,2,3)$, then the third vertex is
Updated On: Jul 7, 2022
- $(3,4,5)$
- $(6,2,3)$
- $(6,3,2)$
- $(6,3,4)$
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The Correct Option is D
Solution and Explanation
Centroid is $\left(\frac{\sum x_{i}}{3}, \frac{\sum y_{i}}{3}, \frac{\sum z_{i}}{3}\right)$
Let the third vertex be $\left(\alpha, \beta, \gamma\right)$.
$\therefore 5=\frac{4+5+\alpha}{3}$ $\Rightarrow\alpha=6$,
$2=\frac{2+1+\beta}{3}$ $\Rightarrow \beta=3, 3=\frac{1+4+\gamma}{3}$
$\Rightarrow \gamma=4$
$\therefore$ The third vertex is $\left(6 , 3 ,4\right)$.
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Concepts Used:
Equation of a Line in Space
In a plane, the equation of a line is given by the popular equation y = m x + C. Let's look at how the equation of a line is written in vector form and Cartesian form.
Vector Equation
Consider a line that passes through a given point, say ‘A’, and the line is parallel to a given vector '\(\vec{b}\)‘. Here, the line ’l' is given to pass through ‘A’, whose position vector is given by '\(\vec{a}\)‘. Now, consider another arbitrary point ’P' on the given line, where the position vector of 'P' is given by '\(\vec{r}\)'.
\(\vec{AP}\)=𝜆\(\vec{b}\)
Also, we can write vector AP in the following manner:
\(\vec{AP}\)=\(\vec{OP}\)–\(\vec{OA}\)
𝜆\(\vec{b}\) =\(\vec{r}\)–\(\vec{a}\)
\(\vec{a}\)=\(\vec{a}\)+𝜆\(\vec{b}\)
\(\vec{b}\)=𝑏1\(\hat{i}\)+𝑏2\(\hat{j}\) +𝑏3\(\hat{k}\)