Question

Line 1 passes through the origin and the point (1,2).
Line 2 passes through the points (1,3) and (2,5).
Line 3 passes through the origin and the point (-2,1).

Which of these are parallel and which are perpendicular?

• A. Line 3 is perpendicular to both Line 1 and Line 2.
• B. Line 2 is perpendicular to both Line 3 and Line 1.
• C. Line 1 is perpendicular to both Line 3 and Line 2.
• D. Line 3 is parallel to both Line 1 and Line 2.

Answer 1

The correct option is A Line 3 is perpendicular to both Line 1 and Line 2.

$\underset{–}{ConceptInvolved:}$

If a line $L$ has two points as $(x_1,y_1)$ and $(x_2,y_2)$, then the slope of the line $L$ would be:
Slope $=\frac{Changeiny}{Changeinx}=\frac{y_2-y_1}{x_2-x_1}$

$\underset{–}{Method:}$

Line 1 passes through the points (0,0) and (1,2). Slope of Line 1 would be:
Slope $=m_1=\frac{2-0}{1-0}=2$

Line 2 passes through the points (1,3) and (2,5). Slope of Line 1 would be:
Slope $=m_2=\frac{5-3}{2-1}=2$

Line 3 passes through the points (0,0) and (-2,1). Slope of Line 1 would be:
Slope $=m_3=\frac{1-0}{-2-0}=\frac{-1}{2}$

Line 1 and Line 2 have the same slope. But slope of Line 3 is negative reciprocal of the slopes of Line 1 and Line 2.

Therefore, Line 3 is perpendicular to both Line 1 and Line 2.