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

The correct option is B Line 3 is perpendicular to both line 1 and line 2.
If a line L passes through (x1,y1) and (x2,y2), then the slope of L is given by:

Slope =RiseRun=Change in yChange in x=y2y1x2x1

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

Slope of line 1 =m1=2010=2

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

Slope of line 2 =m2=5321=2

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

Slope of line 3 =m3=1020=12

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.

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