الرياضيات
The Perpendicular Bisector of a Line Segment
It crosses a line segment at its midpoint at 90°, and every point on it is the same distance from both ends. The compass construction and its use in loci.
Perpendicular bisector
The line that crosses a line segment at its midpoint at 90°, made up of every point that is the same distance from the segment's two ends.
- أين يقابله الطلاب
- Grade 8 or 9 mathematics, in the constructions and loci section of GCSE and IGCSE geometry.
الإجابة باختصار
The perpendicular bisector of a line segment is the line that cuts it exactly in half at 90°. Every point on it is the same distance from the two ends of the segment, which is why loci questions use it for anything that must be equidistant from two fixed points.
مثال
AB = 6 cm → compasses set to any radius over 3 cm, arcs from A and from B
Set the compasses wider than half of AB, draw an arc above and below the line from A, then repeat from B without changing the radius. The two crossing points are each the same distance from A and from B, and the line through them is the perpendicular bisector. Below 3 cm the arcs never reach each other.
Two descriptions of the same line
The name describes what it does to the segment: bisector because it cuts it into two equal halves, perpendicular because it does so at a right angle. Between them those two words pin down exactly one line.
There is a second description that sounds unrelated and is not: it is the set of every point equidistant from the two ends. A point directly above the midpoint is equally far from both, and so is a point far out along the line, because moving along it lengthens the route to each end by the same amount.
Constructions questions use the first description and loci questions use the second, which is why the same line seems to appear in two different topics.
The construction, and why the compasses matter
Four moves: open the compasses to more than half the length of the segment; draw arcs above and below from one end; draw arcs of the same radius from the other end; join the two points where the arcs cross. The radius has to exceed half the length or the arcs cannot reach each other.
Leave the arcs on the page. In a constructions question the arcs are the evidence that compasses were used, and rubbing them out to make the diagram tidy removes the working. A question that says ruler and compasses only will not accept a midpoint measured with a ruler and a right angle drawn with a protractor, even though the line ends up in the same place.
What it is used for
Whenever a question asks for points nearer to one thing than another — the region closer to town A than to town B, the boundary between two mobile masts, where to place a pipe equally far from two wells — the perpendicular bisector is the boundary, and the shading goes on one side of it.
It has a use inside triangles too. Construct the perpendicular bisector of each of the three sides and all three meet at a single point, the circumcentre, which is the centre of the circle passing through all three corners.
أسئلة شائعة
Why must the compasses be set to more than half the length?
Because the arcs from the two ends have to overlap. If the radius is less than half the segment, each arc stays in its own half of the page and they never cross, so there are no points to join. Any radius above half works, and one a little larger than half gives crossings that are easy to see.
Does the perpendicular bisector pass through the midpoint?
Always — that is what bisector means. The useful consequence is that the construction finds the midpoint for you: the place where the drawn line cuts the segment is the exact middle, without any measuring. That is often quicker and more accurate than reading a ruler, particularly on an awkward length.
What is the difference between a perpendicular bisector and an angle bisector?
A perpendicular bisector halves a line segment and gives points equidistant from two points. An angle bisector halves an angle and gives points equidistant from two lines. Loci questions use both, and the wording tells you which: equidistant from two towns is the first, equidistant from two roads the second.
Can a perpendicular bisector be drawn for a line rather than a segment?
Not usefully. A full line has no ends and no midpoint, so there is nothing to bisect. The construction needs a segment with two definite endpoints, which is why questions always give you a length such as AB rather than an equation of a line.
آخر تحديث
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