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 the following is true?

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

Answer 1

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

If a line L passes through $(x_1,y_1)$ and $(x_2,y_2),$ then the slope of L is given by:

$\text{Slope}=\frac{\text{Rise}}{\text{Run}}=\frac{\text{Change in}y}{\text{Change in}x}=\frac{y_2–y_1}{x_2–x_1}$

Line 1:
Line 1 passes through the points $(0,0)$ and $(1,2).$

$\text{Slope of line 1}=m_1=\frac{2–0}{1–0}=2$

Line 2:
Line 2 passes through the points $(1,3)$ and $(2,5).$

$\text{Slope of line 2}=m_2=\frac{5–3}{2–1}=2$

Line 3:
Line 3 passes through the points $(0,0)$ and $(–2,1).$

$\text{Slope of line 3}=m_3=\frac{1–0}{–2–0}=\frac{–1}{2}$

Line 1 and line 2 have the same slope. Thus, they are not perpendicular to each other. In fact, they are parallel to each other.

But the slope of line 3 is the negative reciprocal of the slopes of both line 1 and line 2.

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