Geometric constructions: perpendicular bisector (video)

High school geometry

• Geometric constructions: congruent angles
• Geometric constructions: parallel line
• Geometric constructions: perpendicular bisector
• Geometric constructions: perpendicular line through a point on the line
• Geometric constructions: perpendicular line through a point not on the line
• Geometric constructions: angle bisector
• Justify constructions
• Congruence FAQ

Geometric constructions: perpendicular bisector

Sal constructs a perpendicular bisector to a given line segment using compass and straightedge. Created by Sal Khan.

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• can someone prove why those 2 circle intersection points create a perpendicular bisector?(21 votes)
• You might have to get a piece of paper and draw this out to see it, but if you draw the line and the 2 circles, and then connect each of the two points of intersection with the center of each circle, you get a rhombus(all the sides are radii and therefore equal). Because the original line & the line created by the intersection are both diagonals of the constructed rhombus, they are perpendicular bisectors to each other (a property of rhombuses is that the diagonals are perpendicular bisectors to each other).(33 votes)
• Can these be made exercises so students can practice the constructions? Right now I only see that they’re videos.(15 votes)
• As with many things on Khan Academy the content is experiencing constant “tweaking” in terms of what resources are available, which videos and exercises are found in a given catagory or sub-catagory, and how you would find and access them. During the last academic year there were Compass Construction exercises that I had my students (I am a Geometry teacher) practice as part of what they needed to do when we covered constructions. As tools appear similar/identical to the tools that are used on the PARCC test it was time well spent. This year I looked and discovered the exercises I used are no longer available 🙁 Hopefully will be coming back someday soon.(2 votes)
• Can someone explain this because I don’t get it(8 votes)
• We want to find a line that is perpendicular to the given line. Just drawing any line through the given line won’t do, we want to find one that is perfectly perpendicular to it. In order to find it, we have to use a number of tools: a compass (which in real life helps us draw perfect circles) and a straight edge (anything that would help you draw a straight line, like using the side of a ruler).
To find the perpendicular line we use two compasses to draw two circles, both on the given line. The places where the two circles meet can hep us draw a perpendicular line. Simply use those two point to draw a line, and that line will be perpendicular to the given line.(13 votes)
• Why is it called a Straightedge? I’ve never heard of it til now.(4 votes)
• Not trying to be smart with you, but it’s because it has a straight edge. While typically you would use a ruler as your straight edge, you could use anything that is long and straight. So the side of a book, or the edge of you desk, or the side of a Monopoly board are all perfectly good straight edges. The main thing to remember is that you are only using it to make lines using points from your construction so you can’t cheat by measuring how long a certain line should be.(13 votes)
• What is a perpendicular bisector used to find?(2 votes)
• It cuts the line in half and can create any number of isosceles triangles along the perpendicular bisector. We use the concept in equilateral triangles also to work with special right triangles. It finds the midpoint of a line segment.(4 votes)
• My question is not about this video per se but generally answering questions regarding compass constructions. I frequently get my answer, it is marked incorrect, I then scroll through all the hints and the correct answer is right on top of mine. Why why why? Also, does anyone know a way of getting the line segments to stop changing length? I create a segment, drag it somewhere else and it “snaps” onto the closest object, thus changing length. Super frustrating.(3 votes)
• This isn’t just you. No matter how I do it, either an equivalent way that produces the same result or the exact way shown in the hints it takes a dozen tries to get it ‘right’. SO FRUSTRATING. Apparently I’m never going to get 100% on geometry.(2 votes)
• can we place the circles anywhere ?(2 votes)
• I am not sure exactly what you mean because the center of the circles will always have to be at the endpoints, so we cannot place them anywhere. If you are talking about the size of the radius It has to be bigger than 1/2 of the line segment, so if you estimate it to be about 3/4 of the line, you will be in good shape. The radius of the circle can be different sizes, but you still have to use the same radius from both endpoints.
Note that for a perpendicular bisector, every point on it will be equidistant from the two endpoints (If I drew two triangles, they would be congruent by SSS).(3 votes)
• Can’t we make the circle smaller, like if AB is too large for my compass? I’ve been told you can make it smaller.(2 votes)
• Yes, but you want it to be much bigger than 1/2 the line (3/4 would be great). If it is too close to 1/2. the two points are harder to distinguish.(2 votes)
• 1:02
why do we need to draw 2 circles, instead of one. each circle has a center can’t the perpendicular line be drawn with it? we don’t need to take 2 symmetrical circles.(2 votes)
• Any point on the perpendicular bisector has to be at an equal distance from the two ends of the line segment. Sal has found two such points, and that is enough for him to draw the line. With only one circle, all points on the circle are at an equal distance from one end of the line segment, but we don’t know which of them are also the same distance from the other end of the line segment.(2 votes)
• Is there a video on how to construct parallel lines?(2 votes)
• No but here is a link to a step by step video to how to create a parallel line http://www.mathopenref.com/constparallel.html(2 votes)

Video transcript

We’re asked to construct a perpendicular bisector of the line segment AB. So the fact that it’s perpendicular means that this line will make a 90-degree angle where it intersects with AB. And it’s going to bisect it, so it’s going to go halfway in between. And I have at my disposal some tools I can put out. I can draw things with a compass, and I can add a straight edge. So let’s try this out. So let me add a compass. And so this is a virtual compass. So in a real compass, it’s one of those little metal things where you can pivot it on one point, and you can draw a circle of any radius. And so here I’m going to draw– I’m going to center it at A. And I’m going to make the radius equal to the length of AB. Now I’m going to add another circle with my compass. And now, I’m going to center it at B and make the radius equal to AB. And now, this gives me two points that I can actually use to draw my perpendicular bisector. If I connected this point and this point, it is going to bisect AB, and it’s also going to be perpendicular. So let’s add a straight edge here, so this is to draw a line. So I’m going to draw a line between that point and that point right over there. And let me scroll down, so you can look at it a little bit clearer. So there you go. That’s my construction. I’ve made a perpendicular bisector for segment AB. Check my answer. We got it right.

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