We will learn how to construct a line perpendicular to another line, called a perpendicular line. Learn the two cases: “from a point on the line” and “from a point outside the line.”
Perpendicular to ℓ Through Point P on Line ℓ
① With P as the center, draw an arc of a suitable radius, and let its intersections with ℓ be A and B.
② With A and B as centers, draw arcs with the same radius (greater than PA), and let their intersection be Q.
③ Draw line PQ → this is the perpendicular.
This procedure is the same as constructing the perpendicular bisector of AB. Since P was chosen as the midpoint of AB, PQ is perpendicular to AB and passes through P.
Perpendicular from Point P Outside Line ℓ to Line ℓ
① With P as the center, draw an arc of a suitable radius, and let its intersections with ℓ be A and B (use a radius greater than the distance from P to ℓ).
② With A and B as centers, draw arcs with the same radius and choose their intersection Q on the opposite side of ℓ.
③ Draw line PQ → this is the perpendicular (its foot is the midpoint of AB).
There is a line ℓ and a point P not on ℓ. State the steps for constructing a perpendicular from P to ℓ.
Solution:
① With P as the center, draw an arc that intersects ℓ at two points A and B.
② From A and B, draw arcs with the same radius on the opposite side of ℓ, and choose their intersection Q.
③ Draw PQ → this is the perpendicular.
Use of Perpendiculars: Distance from a Point to a Line
The shortest distance from a point P not on line ℓ to ℓ is the length of the perpendicular from P to ℓ.
Distance from a point to a line ＝ length of the perpendicular
You want to construct a perpendicular from vertex A of a triangle to its opposite side BC. How should you construct it?
Solution:
Consider line BC and construct a perpendicular from point A to line BC, using the same procedure as constructing a perpendicular from a point outside a line.
The foot of the perpendicular lies on BC or its extension.
- Constructing a perpendicular: basically, create two symmetrical points and connect them.
- From a point on a line: draw an arc centered at the point → create two intersection points → create a symmetric point from them → connect the points.
- From a point outside a line: use an arc centered at the point to intersect the line at two points → create a symmetric point on the opposite side → connect the points.
- The distance from a point to a line is the length of the perpendicular (the shortest distance).
Practice Problems
- State the steps for constructing a perpendicular to ℓ through point A on line ℓ.
- Construct a perpendicular from vertex A of triangle ABC to its opposite side BC. What is this perpendicular called?
- Describe how to find the shortest distance from a point P not on line ℓ to ℓ by construction.
Answers and Explanations
- Answer: With A as the center, draw an arc of a suitable radius and create intersections P and Q with ℓ. From P and Q, draw arcs with the same radius and find their intersection R. Draw AR.
Explanation: This is the perpendicular through point A on ℓ. - Answer: Use an arc centered at A to intersect BC at two points. From those two points, draw arcs with the same radius and create their intersection on the opposite side of BC. Connect it to A. This is the “altitude of the triangle (perpendicular).”
Explanation: The perpendicular drawn from a vertex of a triangle to the opposite side is called its altitude. - Answer: Construct a perpendicular from P to ℓ. The length of that perpendicular, from P to its foot on ℓ, is the shortest distance.
Explanation: The length of the perpendicular is the shortest distance.