when constructing a perpendicular bisector why must the arcs intersect

when constructing a perpendicular bisector why must the arcs intersect

when constructing a perpendicular bisector why must the arcs intersect

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When constructing an angle bisector, why must the arcs intersect?

The arcs must intersect in order to connect to the vertex of the angle.

When constructing an angle bisector, we open our compass to any width, and place the point of the compass on the vertex of the angle. We then construct an arc through both sides of the angle.

Next we move the compass to the point where our constructed arc intersects one side of the angle. We then draw an arc inside of the angle.

Using the compass set to the same width, we move the compass to the point where our first arc intersects the other side of the angle, drawing an arc inside of the angle. This new arc will intersect our previous arc, creating a point.

We then use a straightedge to connect this point to the vertex of the angle, giving us our bisector.

If the two arcs did not intersect, we would not have a point to connect to the vertex.

Remarks:

  1. If we construct the perpendicular bisectors of all three sides of a triangle we obtain the circumcenter of the triangle.
  2. When constructing a perpendicular bisector why must the compass opening be greater than 1/2 because otherwise the circular arcs drawn using the compass will not meet each other. If we do not obtain intersecting points then we cannot draw the perpendicular bisector.
  3. Constructing a perpendicular bisector similar to constructing an angle bisector in the sense that in both cases we wish to divide something (a line and an angle respectively) into two equal halves. The word “bisect” literally means dividing something into two equal parts.
when constructing a perpendicular bisector why must the arcs intersect
when constructing a perpendicular bisector why must the arcs intersect

What is Perpendicular Bisector?

A perpendicular bisector is a line that bisects a line segment in two equal parts and makes an angle of 90 degrees at the point of intersection. In other words, we can say that a perpendicular bisector divides a line segment at its midpoint making an angle of 90 degrees. Let us go through the formal definition of it in the next section to understand its meaning in a better way.

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Perpendicular Bisector Definition

A perpendicular bisector is defined as a line or a line segment that divides a given line segment into two parts of equal measurement. ‘Bisect’ is the term used to describe dividing equally. Perpendicular bisectors intersect the line segment that they bisect and make four angles of 90° each on both sides. Perpendicular means a line or a line segment making an angle of 90° with another line or line segment. In the figure shown below, the perpendicular bisector bisects the line segment AB into two equal halves.

Steps for Constructing Perpendicular Bisector

Follow the steps below to construct a perpendicular bisector of a line segment.

  • Step 1: Draw a line segment XY of any suitable length.
  • Step 2: Take a compass, and with X as the center and with more than half of the line segment XY as width, draw arcs above and below the line segment.
  • Step 3: Repeat the same step with Y as the center.
  • Step 4: Label the points of intersection as ‘P’ and ‘Q’.
  • Step 5: Join the points ‘P’ and ‘Q’. The point at which the perpendicular bisector PQ intersects the line segment XY is its midpoint. Label it as ‘O’.
why must the arcs intersect
why must the arcs intersect

Perpendicular Bisector Properties

Perpendicular bisectors can bisect a line segment or a line or the sides of a triangle. The important properties of a perpendicular bisector are listed below.

Perpendicular bisector,

  • Divides a line segment or a line into two congruent segments.
  • Divides the sides of a triangle into congruent parts.
  • They make an angle of 90° with the line that is being bisected.
  • They intersect the line segment exactly at its midpoint.
  • The point of intersection of the perpendicular bisectors in a triangle is called its circumcenter.
  • In an acute triangle, they meet inside a triangle, in an obtuse triangle they meet outside the triangle, and in right triangles, they meet at the hypotenuse.
  • Any point on the perpendicular bisector is equidistant from both the ends of the segment that they bisect.
  • Can be only one in number for a given line segment.
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Important Notes on Perpendicular Bisector

  • A perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point.
  • The perpendicular bisector of a triangle is considered to be a line segment that bisects the sides of a triangle and is perpendicular to the sides.
  • Perpendicular bisector on a line segment can be constructed easily using a ruler and a compass.

MORE TO READ:

How do you draw a perpendicular bisector?

Drawing perpendicular bisector for a line: Place the sharp end of a pair of compasses at one end of the line, and open it to just over half of the line. Draw an arc which must intersect the line in the position described. Then put the sharp end at the other of the line and, keeping the compassing at the same length, draw another arc which intersects the first one twice and also the line. Then draw a straight line through the two places where the arcs intersect. This line is the perpendicular bisector.

why must the arcs intersect
why must the arcs intersect

Drawing perpendicular bisector of angle: Places the sharp end of the compass at the point of the angle and, after having opened it arbitraily wide, draw an arc which intersects the two lines meeting to form the angle each once in the said position. Then remove the compass and, always keeping it opened at the SAME length, place the sharp end at each of the two places where the previous arc cuts each of the two lines meeting to form the angle. In this position with the described length, draw a small arc at each of the said places, which should cross each other. Draw a straight line from the point of the angle to this crossing. This should be the bisector of the angle.

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The measure of a tangent-tangent angle is half the difference of the measures of the intercepted arcs?

True


How do you find midpoint of a segment?

Take a compass, extend it about 3/4 of the length of the segment. Then from one end of the segment, draw a 180 degree arc. From the other end draw another arc. Connect the points where the arcs intersect. Where the line intersects with the segment is the midpoint of the segment. That is how you bisect a segment to find the midpoint – geometrically.


Measures of minor arcs equal measures of inscribed angles?

lol. your on odyssey ware


How do you draw a spherical triangle where the sum of the angles is 190 degrees?

You could draw two arcs from the North pole to the equator, with a 10 degree separation. The two arcs and the equator would form a 190 degree spherical triangle.

 

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