Properties of a perpendicular bisector
In geometry, the property of a perpendicular bisector is the relationship between two lines that meet at a right angle. This perpendicular bisector property is applicable in other geometric objects. In a perpendicular bisector, a line is perpendicular to another line if both lines intersect at a right angle. In short, the first line is perpendicular to the second line if both lines meet and at that point where they meet, one side of the line is divided by the second line into two congruent angles. The aspect of a perpendicular bisector can be symmetric implying that if the first line is perpendicular to the second line, then that second line is also perpendicular to the first line. A perpendicular bisector is used to create right angles at the middle of a segment. Every point in a perpendicular bisector is equidistant from the endpoints of a given segment. The point where the perpendicular bisector of a triangle meets is equidistant from the triangle vertices. In perpendicular bisector geometry, a very important line, segment or ray that can help to proof congruence is known as the angle bisector. And a good example of an angle bisector is the perpendicular bisector. A perpendicular bisector is a special kind of segment, line or ray. Other properties of a perpendicular bisector are:
- It passes through a segment at its midpoint
- If a line L acts as a perpendicular bisector of a line AB and L divides AB into 2 equal parts at a right angle. C is the midpoint of the line AB and the line L intersects the line segment to form a perpendicular bisector.
- The line which is perpendicular to another straight line and it bisects it; it is the perpendicular bisector of that line.
- Perpendicular lines intersect each other at a right angle.
- A curve or a line which divides or bisects a line or an angle into two equal parts is known as a bisector.
Two theorems for the characteristics of a perpendicular bisector are derived to obtain the solutions for geometric computations. One of them is the perpendicular bisector theorem. This perpendicular bisector theorem states that if a point is lying on a perpendicular bisector of a line, then it is equidistant from the line end points. The converse is also true. If a point is equidistant from the line’s endpoints, then it is lying on the perpendicular bisector of a line. These theorems simply show the existence of locus points which form a perpendicular bisector. These points have the same length from the endpoints of the line which meets in the middle at right angles on that line. It is important to also note the difference between an angle bisector and a perpendicular bisector. An angle bisector splits an angle into two congruent angles while a perpendicular bisector forms a right angle at the middle of a segment.
How to construct a bisector perpendicular to a line
To construct a perpendicular bisector of a line AB from a point P, you begin by constructing a circle with a center P to form point A and B on a line AB which is of equal distance from P. You will then construct circles with centers A and B which have an equal radius. Let R and Q be the points where these two circles intersect. The last step will be to join R and Q to form a perpendicular bisector PQ.
The construction of a perpendicular bisector of a line is a very common plane geometry. Another method of constructing a perpendicular bisector begins by measuring the length which you will then divide into two and note the midpoint. You can also construct a perpendicular bisector without any measurement by using a compass and a ruler or a straight edge. To construct a perpendicular bisector of a line AB with a compass and a ruler, start by stretching the compass to a length that is more than half the length of AB. With the sharp end of the compass at A, mark an arc at the top and at the bottom of the AB. Without altering the compass length, put the sharp end at point B also and mark arcs at the bottom and at the top of line AB which will intersect with the previous arcs made with the compass centre A. Then connect the two points where the arcs meet with a straight line. That line is the perpendicular bisector of Line AB and P is its midpoint. This method of constructing a perpendicular bisector can also be used to construct an isosceles triangle.
For a perpendicular bisector of a chord, if you have a line running through the centre of a circle which is perpendicular to a chord, then the line becomes the perpendicular bisector of that chord since it cuts it into two equal portions. If the line is running through the centre of the circle which cuts the chord into two equal portions, then that line is the perpendicular bisector of that chord. And if you have a line which is a perpendicular bisector of a chord or cuts that chord into two equal parts and intersects the chord at 90 degrees, then that line will pass through the midpoint of the circle.
Equations and calculations
To find the perpendicular bisector equation of two given points, all you need is to determine their middle point and the negative reciprocal and then plug the answers into an equation for a line in slope intercept form. To find the perpendicular bisector of two points, start by gathering information. Know the middle point of the two given points by using the midpoint formula [(x1+x2)/2,(y1+y2)/2]. This formula in a perpendicular bisector equation simply finds the average of the y and x coordinates of the two points which you intend to find their midpoint coordinates. If (x1, y1) has the coordinates (2, 5) and (x2, Y2) has the coordinates (8, 3) the midpoint will be found by [(2+8)/2, (5+3)/2]= (5, 4) the midpoint is therefore (5, 4).
The next step in finding the perpendicular bisector equation of two points is finding the slope of the points. To calculate the slope, use the formula (y2-y1)/(x2-x1). The slope of a line is the measure of its vertical change over the horizontal change. The slope of points (2, 5) and (8, 3) is therefore (3-5)/(8-2)= -2/6= -1/3. The slope or gradient of the perpendicular bisector equation is negative 1/3. To find the negative reciprocal of the slope of these two points, you take the reciprocal of the slope and change its sign. You may take the reciprocal of a number by simply flipping the y and x coordinates. For example, the reciprocal of ½ is -2/1 or just negative 2 and the reciprocal of negative 4 is ¼. Now the negative reciprocal of negative 1/3 is just 3 because 3/1 is the reciprocal of 1/3 and the sign changes from negative to positive.
The next step in determining the perpendicular bisector equation is solving the equation of the line. Write down the line equation in a slope-intercept form which is y=mx+b whereby the letters x and y will represent the x and y coordinates and the letter m represents the gradient or the slope of that line and b is the y intercept. A y intercept is where the line cuts the x axis. Once you have written this perpendicular bisector equation format, you can start finding the equation of the perpendicular bisector of the two points. Plug in the negative reciprocal of the initial slope into the equation. The negative reciprocal of the slope was 3. The m in the equation will be replaced by 3 thus y=3x+b. Insert the midpoints into the line. The midpoint was (5, 4) and since the perpendicular bisector runs through the middle point of a line, insert the midpoint coordinates into the equation of the line. The x and the y in the equation will be replaced by 5 and 4 thus y=mx+b will be 4=3(5) +b=4=15+b.
The last step in finding the perpendicular bisector equation for two given points is solving for the intercept. Since you already have three variables in the equation, it is very easy to find b which is the y intercept of the line. Subtract 15 from both sides of the equation 4=15+b. -11=b so b is negative 11. Now write down the equation of the perpendicular bisector. To write the equation, you just plug in the slope of the line which is 3 and the y intercept which is (-11) into the line equation in slope intercept form. The x and y will remain as letters because this particular perpendicular bisector equation will help you to calculate any coordinate in the line by plugging in either of the x or y coordinate. Y=mx +b becomes y=3x-11. The perpendicular bisector equation for points (2, 5) and (8, 3) is y=3x-11.
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