Triangle formed by three lines whose direction ratios are −1,1,1,1,2,0,1,1,0 is

# Triangle formed by three lines whose direction ratios are $〈-1,1,1〉,〈1,\sqrt{2},0〉,〈1,1,0〉$ is

1. A

Acute angled triangle

2. B

Obtuse angled triangle

3. C

right angled triangle

4. D

coincident lines

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### Solution:

Suppose that the given lines form a triangle $ABC$

Hence,

$\begin{array}{c}\mathrm{cos}A=\frac{-1+\sqrt{2}}{\sqrt{1+1+1}\sqrt{1+2}}=\frac{\sqrt{2}-1}{3}\\ \mathrm{cos}B=\frac{1+\sqrt{2}}{\sqrt{1+2}\sqrt{1+1}}=\frac{\sqrt{2}+1}{\sqrt{6}}\\ \mathrm{cos}C=\frac{-1+1}{\sqrt{1+1}\sqrt{1+1+1}}=0\end{array}$

Since $\mathrm{cos}C=0$,the measure one angle in the triangle is right angle, so that the triangle formed by the given lines is right angled triangle.

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