# Lesson Explainer: Geometric Constructions: Perpendicular Lines Mathematics

In this explainer, we will learn how to construct, using a ruler and a pair of compasses, the perpendicular to a given line from or at a given point and the perpendicular bisector of a line segment.

We first note that for any line , there is a unique line that is perpendicular to that passes through a given point . We can construct this line with a compass and a straight edge by using our knowledge of circles and triangles.

We start by setting the point of our compass at , and then we can use any size for the radius. Showing only the arcs that intersect gives us the following.

We can call the points of intersection between this circle and   and respectively. We note that since these are both radii of the same circle, so we can say that is equidistant from and . If we then set the radius of the compass to be larger than , we can trace circles (or just their arcs) centered at and such that they intersect twice. We will call the point of intersection above  , as shown.

We then note that , as they are radii of congruent circles. We can add these lines along with onto the diagram.

We now note that triangles and have congruent sides, so by the SSS criterion, they are congruent triangles. In particular, this means that . For two congruent angles to make up a straight angle, their measures must sum to , so they are both right angles.

Hence, we have constructed , which is perpendicular to passing through .

We can follow this process to find the perpendicular line passing through any given point on a line.

### How To: Constructing the Perpendicular Line through a Given Point on the Line

To find the perpendicular line to passing through , we use the following steps:

1. We start by setting the point of our compass at and then trace a circle intersecting twice. We can label these points and .
2. We then set the radius of the compass to be larger than and we can trace arcs of circles centered at and such that they intersect. We will call the point of intersection above  , as shown.
3. We have that .

This allows us to find the perpendicular line passing through any given point on a line. However, there is a special case of this that splits into two equal parts called the perpendicular bisector. We define this formally as follows.

### Definition: Bisectors and Perpendicular Bisectors

To bisect a line segment means to separate it into two equal parts.

In particular, the perpendicular bisector of a line segment is the line that separates the line segment into two equal parts and meets the line segment at right angles.

We can follow a similar process to construct the perpendicular bisector of any line segment . We start by setting the radius of the compass to be greater than and then sketching arcs of circles of this radius centered at and . These will intersect at two points that we will label and , as shown.

Connecting and with a straight line and adding in the radii gives us the following.

We note that and have three congruent sides, so by the SSS criterion, they are congruent. Therefore, their corresponding angles are congruent. So, . We can add this onto the diagram along with labeling the point of intersection between and  .

We then see that and share two congruent sides and the included angles are also congruent. Thus, by the SAS criterion.

Hence, their corresponding sides are all congruent, and so . We also note that and they sum to make a straight angle, so they must be right angles. It is also worth noting that this proves the diagonals of a rhombus meet at right angles.

This means that is the perpendicular bisector of . We can follow this process in general to construct the perpendicular bisector of any line segment.

### How To: Constructing the Perpendicular Bisector of a Line Segment

To find the perpendicular bisector to , we use the following steps:

1. Set the radius of the compass to be greater than .
2. Trace arcs of two circles of this radius centered at and . These will intersect at two points that we will name and .
3. is then the perpendicular bisector of .

Let’s now see an example of determining the relationship a geometric construction illustrates.

### Example 1: Constructing the Perpendicular Bisector of a Line Segment

Which of the following does the figure illustrate?

What does the following figure illustrate

1. A perpendicular to a straight line originating from it
2. A bisector of an angle
3. A perpendicular bisector of a line segment
4. A straight line parallel to another line
5. A perpendicular from a point lying outside a straight line

We can answer this question by recalling that the perpendicular bisector of a line segment is constructed by the line segment between the intersection of circles centered at and . This is exactly what is pictured, so the answer is option C: a perpendicular bisector of a line segment.

In our next example, we will see another question on determining the relationship a given geometric construction illustrates.

### Example 2: Constructing the Perpendicular to a Line from a Point on the Line

What does the following figure illustrate?

1. A bisector of an angle
2. A perpendicular to a straight line originating from it
3. A perpendicular from a point lying outside a straight line
4. A straight line parallel to another line
5. A bisector of a line segment

We can answer this question by recalling that the perpendicular bisector of a line segment through a point on the line is constructed by first tracing a circle at to find two points on equidistant from —say and . Then, we trace circles at and to find a second point equidistant from and —say . We then recall that the line segment is perpendicular to .

This is exactly what is pictured, so the answer is option B: a perpendicular to a straight line originating from it.

Before we move on to the example, there is one more construction we need to consider. We have seen how to find the perpendicular line to through a point on the line, but what if does not lie on the line?

We start by tracing a circle centered at that intersects twice; we label these points and .

We note that is equidistant from and , since and are radii of the circle centered at . We can find another point equidistant from and by tracing circles of the same radius centered at each point that intersect at a point , as shown.

Since we kept the radii of the circles the same, , adding these congruencies and the line segment to the diagram gives us the following.

This is the same construction we had previously, where and are diagonals of a rhombus. Thus, we can conclude they meet at right angles.

We can follow this process in general to construct the line perpendicular to any line through a point not on the line.

### How To: Constructing the Perpendicular of a Line through a Point Not on the Line

To find the perpendicular to passing through , we use the following steps:

1. Trace a circle centered at that intersects twice; label these points and .
2. Trace circles centered at and that intersect; label the point of intersection .
3. is then perpendicular to .

Let’s now see an example of using this construction to identify which construction represents drawing a perpendicular from a point lying outside a straight line.

### Example 3: Identifying the Construction of the Perpendicular to a Line from a Point Lying outside the Line

Which of the following constructions represents drawing a perpendicular from a point lying outside a straight line?

We recall that to find a perpendicular from a point lying outside the straight line , we first trace a circle at that intersects at two distinct points—say and . If we then trace circles centered at and that intersect at a point , then .

We can see that this option E.

It is worth noting that the other options give different geometric constructions. Option A is a construction of a perpendicular bisector of a line segment. Option B finds the perpendicular line to a straight line passing through a point on the line. Option C is a bisector; however, it is not perpendicular. And option D is beyond the scope of this explainer, but it is a construction of an angle bisector.

So, the answer is option E.

Before we move on to our next example, we can show a few useful properties of the perpendicular bisector of a line segment.

Consider the following line segment with perpendicular bisector , as shown.

We can first show that every point on is equidistant from and . To do this, let’s consider any point . We can construct triangle as shown.

We note that triangles and share two congruent sides and the included angles are both right angles. So, they are congruent. Hence, .

We now note that since every point on the perpendicular bisector is equidistant from both and , the perpendicular bisector of a line is also its axis of symmetry.

Finally, we can also show that all points equidistant from and lie on the perpendicular bisector. If we let be any point that is equidistant from and , then either is coincident with or triangles and are congruent by the SAS criterion.

Let’s now see an example of determining which quadrilateral is given by a geometric construction.

### Example 4: Linking the Construction of the Perpendicular Bisector of a Line Segment with the Properties of a Rhombus

Draw the given figure and connect the points . What does that figure represent?

We first note that and are the points of intersection between two circles of radius centered at and . We recall that this means that is the perpendicular bisector of . If we then add these right angles and quadrilateral onto the diagram and label the center point , we get the following.

We note that , , , and all have two congruent sides and the included angles are all right angles. So, all four triangles are congruent.

Thus, their corresponding sides are all congruent. This means all four sides of the quadrilateral are equal in length. We note that this is not necessarily a square, since we do not know if and have the same length. Hence, we can conclude that the quadrilateral is a rhombus.

Before we move on to our final example, we are equipped to discuss the concept of the shortest distance between a point and a line. Let’s say we have a line and a point and we want to determine the shortest distance from the point to the line.

We can first note that we can find the distance between any point on the line, , and by finding the length of .

However, if we also sketch the perpendicular line from to , we can note that is the hypotenuse of a right triangle and so is longer than the perpendicular distance.

Hence, the perpendicular distance between a point and a line is the shortest distance.

In our final example, we will use this property to identify a geometric construction of the shortest distance between a point and a line.

### Example 5: Identifying the Shortest Distance from a Point to a Line

In the following figure, what is the shortest distance between point and ?

We start by recalling that the shortest distance between a point and a line is the perpendicular distance. Thus, we need to find the perpendicular distance from point to .

We note that the line is intersected by two circular arcs from a circle centered at . Therefore, if we call the points of intersection between the line and this circle and , then we know that , since these are radii of the same circle; it means that is equidistant from and .

We then note that lies on circular arcs of the same radius centered at and . So, , which means that is equidistant from and .

Hence, both and are equidistant from and , so they both lie on the perpendicular bisector of . Therefore, the point of intersection between and is the perpendicular bisector of ; this is labeled .

Thus, and , so is the shortest distance from to .

Let’s finish by recapping some of the important points from this explainer.

### Key Points

• To find the perpendicular line to passing through , we use the following steps:
1. We start by setting the point of our compass at and then trace a circle intersecting twice. We can label these points and .
2. We then set the radius of the compass to be larger than , and we can trace circles centered at and such that they intersect at a point we will call .
3. We then have that .
• The perpendicular bisector of a line segment is the line that separates the line segment into two equal parts and meets the line segment at right angles. Every point on the perpendicular bisector of is equidistant from and . All points equidistant from and lie on the perpendicular bisector of . The perpendicular bisector of a line segment is its axis of symmetry.
• To find the perpendicular bisector of , we use the following steps:
1. Set the radius of the compass to be greater than .
2. Sketch circles of this radius centered at and . These will intersect at two points that we will name and .
3. is then the perpendicular bisector of .
• To find the perpendicular to passing through , we use the following steps:
1. Trace a circle centered at that intersects twice; label these points and .
2. Trace circles centered at and that intersect; label the point of intersection .
3. is then perpendicular to .
• The shortest distance between a line and a point is the perpendicular distance.