High School

Let \( l_1, l_2, \) and \( l_3 \) be three lines such that \( l_1 \) is perpendicular to \( l_2 \) and \( l_3 \) is perpendicular to both \( l_1 \) and \( l_2 \).

Which point lies on \( l_3 \)?

A) Intersection of \( l_1 \) and \( l_2 \)
B) Midpoint of the segment joining the intersection of \( l_1 \) and \( l_2 \)
C) On the y-axis
D) On the x-axis

Answer :

Final answer:

The point which lies on l3, which is perpendicular to both l1 and l2, is the intersection of l1 and l2. option (A) is correct.

Explanation:

Considering that l1 is perpendicular to l2 and l3 is perpendicular to both l1 and l2, it follows that l3 must intersect l1 and l2 at a single point. This point of intersection is the only point common to all three lines since l1 and l2 intersect at exactly one point by being perpendicular to each other. According to the theorem that states, 'If two lines are perpendicular to a third, points on either equidistant from the third are equidistant from the other,' the point of intersection of l1 and l2 will be the pole of l3. As such, the correct answer to the student's question is A) Intersection of l1 and l2.

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