Bisect
Imagine a lovely cake with delicious frosting that needs to be divided at a birthday. The person cutting the cake will not divide the cake into multiple pieces, as it will create quite the mess.
Instead, the person will first divide the cake into two equal halves. This is called bisection and it is an important part of geometry and how we study angles.
In this lesson, we will learn how to bisect a segment, how to bisect lines, and the rules that are applied while bisecting angles.
Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
What is Bisect?
Bisect: Definition
Bisect means to cut or divide into two equal parts.
Bisecting a Line Segment
Let's do an activity to understand the meaning of bisecting a line segment.
Fold a sheet of paper and let \(\overline{AB}\) be the line of fold.
Put an ink-dot on any one side of \(\overline{AB}\) and name it as \(X\)
Can you find the mirror image of \(X\)?
Name the mirror image of \(X\) as \(X'\).
Let \(\overline{AB}\) and \(\overline{XX'}\) intersect at \(O\).
Is \(\overline{OX}\) and \(\overline{OX'}\) equal?
This means \(\overline{AB}\) divides \(\overline{XX'}\) into two equal parts.
So, here \(\overline{AB}\) bisects \(\overline{XX'}\) and \(\overline{AB}\) is a bisector of \(\overline{XX'}\).
Use the simulation below to learn how to bisect a line segment using a ruler and a compass.
Bisecting an Angle
Let's do another activity to understand how to bisect an angle.
Mark a point \(O\) on a paper.
Taking \(O\) as the initial point, draw two rays \(\overline{OA}\) and \(\overline{OB}\)
You will get \(\angle AOB\)
Now, fold the sheet through \(O\) such that the rays \(\overline{OA}\) and \(\overline{OB}\) overlap each other.
Name the line of the crease on the paper as \(\overline{OC}\)
Can you measure the angles \(\angle AOC\) and \(\angle BOC\)? Are the measures equal?
Yes, both the angles are equal and \(OC\) is the angle bisector of \(\angle AOB\).
Use the simulation below to learn the way to bisect an angle using a ruler and a compass.
- A ray \(BD\) bisects an angle \(\angle ABC\). If \(\angle ABC=120^{\circ}\), what would be the measure of the angles \(\angle ABD\) and \(\angle DBC\)?
- Find the coordinates of the midpoint of the line segment \(\overline{AB}\) joining the points \((-5,3)\) and \((3, 2)\).
Bisecting a Shape
Some shapes can also be bisected.
Look at the shapes shown below.
A line segment bisects each shape into two equal parts.
- Bisect means to cut or divide something into two equal parts.
- You can use a compass and a ruler to bisect a line segment or an angle.
- The bisector of a line segment is called a perpendicular bisector.
Solved Examples
| Example 1 |
Ryan is flying a kite.
The kite has two angles bisected as shown below.
Can you find the measure of the angles \(\angle EKI\) and \(\angle ITE\)?
Solution
The angles \(\angle EKI\) and \(\angle ITE\) are bisected by the line \(KT\)
\(KT\) divides the angles \(\angle EKI\) and \(\angle ITE\) in two equal angles respectively.
Thus,
\(\angle EKI=2\times 45^{\circ}=90^{\circ}\) and
\(\angle ITE=2\times 27^{\circ}=54^{\circ}\)
| \(\therefore\) \(\angle EKI=90^{\circ}\) and \(\angle ITE=54^{\circ}\). |
| Example 2 |
In the diagram shown below, \(RQ\) bisects the angle \(\angle PRS\).
Can you find the value of \(x\)?
Solution:
Since \(RQ\) bisects the angle \(\angle PRS\),
\(\angle PRQ=\angle QRS\)
Using this information, we can solve for \(x\).
\[\begin{align}\angle PRQ&=\angle QRS\\x+40&=3x-20\\40+20&=3x-x\\2x&=60\\x&=30\end{align}\]
| \(\therefore\) The value of \(x\) is \(\ 90^{\circ}\) |
Interactive Questions
Here are few activities for you to practice. Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about Bisect with the simulations and practice questions. Now you will be able to easily solve problems and understand bisect definition, bisect symbol, bisect geometry definition, bisect a segment, bisecting lines, and bisecting angles.
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Frequently Asked Questions (FAQs)
1. How to bisect a line?
Do the following steps to draw a bisector of a line.
- Draw a line segment \(\overline{AB}\) on a paper.
- Taking \(A\) as a center, draw a circle. Just make sure that the radius of the circle should be more than half of the line segment \(\overline{AB}\).
- Now, taking \(B\) as a center and with the same radius, draw another circle. Let this circle intersect the previous circle at the points \(C\) and \(D\).
- Join the points \(C\) and \(D\).
The line \(\overline{CD}\) is the bisector of the line \( \overline{AB} \).
2. What happens when you bisect an angle?
When we bisect an angle, the angle is divided into two equal smaller angles.
3. How do you bisect an angle of 60 degrees?
Let's consider an angle of measure \(60^{\circ}\)
Do the following steps to draw a bisector of an angle.
- Keeping the sharp end of your compass at \(B\), draw an arc on BC and mark it as \(S\).
- Similarly, keeping the sharp end of your compass at \(B\), draw an arc on AB and mark it as \(T\).
- Now, keeping the sharp end of your compass at \(S\), draw an arc within \(AB\) and \(BC\)
- Repeat the third step at \(T\)
- Join the point \(B\) and the intersection of the two arcs.
- The line is the angle bisector of \(\angle{ABC}\).
The line is the angle bisector of \(\angle{ABC}=60^{\circ}\).
4. How do you bisect an angle of 45 degrees?
Assume you have constructed an angle of measure \(45^{\circ}\)
Do the following steps to draw a bisector of this angle.
- Keeping the sharp end of your compass at \(B\), draw an arc on BC and mark it as \(S\).
- Similarly, keeping the sharp end of your compass at \(B\), draw an arc on AB and mark it as \(T\).
- Now, keeping the sharp end of your compass at \(S\), draw an arc within \(AB\) and \(BC\)
- Repeat the third step at \(T\)
- Join the point \(B\) and the intersection of the two arcs.
- The line is the angle bisector of \(\angle{ABC}\).
The line is the angle bisector of \(\angle{ABC}=45^{\circ}\).